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5. De Gruyter Series in Nonlinear Analysis
and Applications 14
Editor in Chief
Jürgen Appell, Würzburg, Germany
Editors
Catherine Bandle, Basel, Switzerland
Alain Bensoussan, Richardson, Texas, USA
Avner Friedman, Columbus, Ohio, USA
Karl-Heinz Hoffmann, Munich, Germany
Mikio Kato, Kitakyushu, Japan
Umberto Mosco, Worcester, Massachusetts, USA
Louis Nirenberg, New York, USA
Boris N. Sadovsky, Voronezh, Russia
Alfonso Vignoli, Rome, Italy
Katrin Wendland, Freiburg, Germany

7. Luboš Pick
Alois Kufner
Oldˇrich John
Svatopluk Fuˇcík
FunctionSpaces
Volume 1
2
nd
Revised and Extended Edition
De Gruyter

8. Mathematics Subject Classification 2010:46E30, 46E35, 46E05, 47G10, 26D10, 26D15,
46B70, 46B42, 46B10.
ISBN 978-3-11-025041-1
e-ISBN 978-3-11-025042-8
ISSN 0941-813X
Library of Congress Cataloging−in−Publication Data
A CIP catalog record for this book has been applied for at the Library of Congress.
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available in the Internet at http://dnb.dnb.de.
© 2013 Walter de Gruyter GmbH, Berlin/Boston
Typesetting: le-tex publishing services GmbH, Leipzig
Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen
1Printed on acid-free paper
Printed in Germany
www.degruyter.com

9. Preface
The book “Function Spaces” [126],published in 1977 by Academia Publishing House
of the Czechoslovak Academy of Sciences in Prague and the Noordhoff International
Publishing in Leyden, proved over several decades to be a useful tool for specialists
working in many different areas of mathematics and its applications. It has, though,
for quite some time been unavailable. Since the 1970s, many other books dedicated
to the study of function spaces and related topics have appeared. Nevertheless, we
saw signs that a new edition of this book could be useful, which of course would be
revised according to the rapid development in the field of function spaces over the
past 35 years and upgraded in part by a number of new results.
The current book is an attempt to make a step in this direction. Thanks to the
effort spent by the de Gruyter Publishing House, the three authors signed below took
upon the task. They used as their point of departure the initial book, thus, the current
version now has four authors.
It turned out during the preparation of the material for the new edition that the
upgraded text is too long for a single monograph. Consequently, we decided to split
the material into two volumes.
The first volume is devoted to the study of function spaces, based on intrinsic prop-
erties of a function such as its size, continuity, smoothness, various forms of control
over the mean oscillation, and so on. The second volume will be dedicated to the study
of function spaces of Sobolev type, in which the key notion is the weak derivative of
a function of several variables.
During almost a century of their existence, Lebesgue spaces have constantly played
a primary role in analysis. However, it has been known almost from the very begin-
ning that the Lebesgue scale is not sufficiently general to provide a satisfactory de-
scription of fine properties of functions required by practical tasks. This was noted
during the early 1920s by Kolmogorov, Zygmund, Titchmarsh and others, mostly in
connection with research of properties of operators on function spaces. Thus, natu-
rally, during the first half of the twentieth century, new fine scales of function spaces
have been introduced. The efforts of Young, Orlicz, Hardy, Littlewood, Zygmund,
Halperin, Köthe, Marcinkiewicz, Lorentz, Luxemburg, Morrey, Campanato and many
others resulted in the development of a powerful and qualitatively new mathematical
discipline of function spaces.
This text is intended to be a motivated introduction to thesubject of function spaces.
It contains important basic information on various kinds of function spaces such as
their functional-analytic or measure-theoretic properties, as well as their important

10. vi Preface
characteristics such as mutual embeddings, duality relations and so on. Hence, it can
be considered as a reference book and a pointer to other sources.
Summary of the text
The text opens with Chapter 1, which has a purely preliminary character. It con-
tains basic information about vector spaces, topological, metric, normed and modular
spaces as well as important ingredients from classical functional analysis. In com-
parison with the first edition, this chapter was enlarged considerably. In particular,
a thorough treatment of the theory of measure and integral was added. There are of
course many possible sources available for this purpose. We mostly use the book
of Rana [185], where the interested reader will find detailed proofs and many further
details. In addition, some new material was added on topological spaces and modular
spaces.
In Chapter 2 we present just a very short elementary discussion aboutspaces of
continuous and smooth functions, including Hölderand Lipschitz spaces.
Chapter 3 contains the material concerning Lebesgue spaces from the first edition,
only slightly modified and upgraded. Some parts (for instance Section 3.11, devoted to
the study of weighted Hardy inequalities) are completely new. A few basic facts about
sequence spaces were added, and the topic of modes of convergence was reworked.
In Chapter 4 we present the study of basic properties of Orlicz spaces. Again,
this chapter as compared to the first edition is only slightly modified. For traditional
reasons, the functions studied in Chapters 3 and 4 are assumed to act on an open subset
of the Euclidean space. The main reason for this restriction is the further use of these
spaces in Sobolev-type spaces, which will be dealt with inthe second volume. For the
study of these spaces themselves, of course, such restriction is not necessary. In fact,
it is later reduced in the frame of the study of general Banach function spaces from
Chapter 6 onwards, noting that Lebesgue and Orlicz spaces are particular examples
of rearrangement-invariant (r.i.) Banach function spaces.
Chapter 5 contains elementary properties of Morrey- and Campanato-type spaces
in which the mean oscillation of a function is measured. The functions in this chapter
are assumed to act on nice domains in the Euclidean space. This chapter is practically
unchanged compared to the first edition.
In Chapter 6 we develop the basic theory of Banach function norms and Banach
function spaces, which was in some sense a culmination of efforts to cover Orlicz
spaces with other types of spacesunder a common theme, performed from the 1930s
to the 1950s by Orlicz, Lorentz, Luxemburg, Zaanen, Köthe, Halperin and others.
This material gradually appeared mostly in works of the mentioned authors. The
systematic treatment of this topic can also be found in the following books: Luxem-
burg and Zaanen [142] or Bennett and Sharpley [14]. Here (except for some minor
additions and changes) we follow the excellent exposition of this subject from [14,
Chapter 1] almost verbatim (even though our ultimate goal is slightly different, as we
are not so much aimed towards interpolation theory).

11. Preface vii
In Chapter 7 we turn our attention to function spaces in which the norm is purely
determined by the size of a given function. Hence, we study the distribution func-
tion and the nonincreasing rearrangement, and develop the resulting structure, the
so-called rearrangement-invariant (r.i.) function spaces. This stuff can be in some
sense traced back to as far as to the 1880s results of Steiner [214], but its first system-
atic treatment was done only in the 1930s in the work of Hardy, Littlewood and Pólya
[101]. Here we once again mostly follow the book by Bennett and Sharpley [14]. The
only significant addition is Section 7.11 in which more recent material concerning
the important relation between function spaces called almost-compact embedding is
studied in great detail. Our main source in this section is [206].
The knowledge of the notion of the nonincreasing rearrangement of a function now
enables us to construct new scales of function spaces that could not exist without
it. We have seen that, for example, Lebesgue and Orlicz spaces just happen to be
rearrangement-invariant Banach function spaces, but for their initial definition we
did not need to know this at all. In subsequent chapters however we study func-
tion spaces for whose definition the nonincreasing rearrangementis indispensable.
First such class of function spaces is that of two-parameter Lorentz spaces, whose el-
ementary properties are studied in Chapter 8. These spaces gradually emerged during
the 1950s and 1960s through the efforts of Lorentz, Calderón, Hunt, Peetre, O’Neil,
Weiss, Oaklander and others, mostly in connection with some kind of interpolation.
We cover some of their elementary properties, embedding characteristics and duality
relations.
In the subsequent two chapters we turn our attention to some of the most inter-
esting generalizations of the two-parameter Lorentz spaces. In Chapter 9, we study
the important scale of theso-called Lorentz–Zygmund spaces, invented in the 1980s
by Bennett, Rudnick and Sharpley, and their generalization to four-parameter spaces,
studied in the 1990s by Edmunds, Gurka, Opic and others in connection with vari-
ous limiting or critical-state problems concerning the action of operators on function
spaces. These spaces turned out to be extremely useful in various extremal problems
concerning Sobolev inequalities as well as limiting properties of operators. They
cover important previously known function classes such as Lebesgue and Lorentz
spaces, Zygmund classes of both logarithmic and exponential type, and also the space
L
1;nI1
, which appeared (under various different symbols) in connection with the
optimal target space for a limiting Sobolev inequality in works by Maz’ya, Hansson,
Brézis–Wainger and others. As we know very well from our own research, these
spaces often arise in various practical tasks. We thus study them here in great de-
tail, concentrating on their embedding and duality characteristics and basic functional
properties. We mostly follow [172] and [73].
In Chapter 10, we investigate the so-called classical Lorentz spaces. These spaces
are currently known to be of three different types (denoted as of typeƒ,andS)and
have been widely studied by many authors. We first concentrate on their basic func-
tional properties such as nontriviality, linearity, normability, quasi-normability, lattice

12. viii Preface
property and so on, and then we focus on their embedding relations. These questions
are highly nontrivial and they require the development of certain new methods some
of which we present in detail. This is the case for instance of the inequalities involving
suprema, which we just quote, or of the modification of the Hardy inequality which
concerns two integral operators rather than just one, which we present in detail. The
spaces of typeSare known to be of great interest because of the gradient inequalities
they govern and their connection to Sobolev and Besov spaces that will be studied in
the second volume of this book. In particular, they contain functions with controlled
nonincreasing rearrangement of mean oscillation. Here we mix classical results with
recent ones scattered in many papers. We follow [33, 34, 35, 56, 89]. At some occa-
sions we do not provide all the details of the proofs because this would increase the
length of the text enormously, and restrict ourselves to the hints and references.
Finally, Chapter 11 is devoted to a brief accountof the generalized Lebesgue spaces
with variable exponent. This topic has become very fashionable in recent years and
there exist entire schools of top scientists investigating all kinds of variable expo-
nent spaces and their generalizations. We restrict ourselves to some basic information
about these spaces, following mostly [122] and [140], and we refer the reader in-
terested in their deep studyto the recent book by Diening, Hästö, Harjulehto and
R˚užiˇcka [63].
Acknowledgments
The main objective of this book is to provide pure mathematicians as well as applied
scientists with a handbook containing a summary of results concerning various types
of function spaces that might be useful for a broad variety of applications. Therefore,
naturally, in most cases we do not claim any originality. We took great effort to give
full credit for all the results appearing in the text to its discoverers but, obviously, it is
almost an impossible task to trace the origins of every detail.
It would be impossible to list all the authors, colleagues and friends who have
influenced us in preparation of the second edition of the text.
The exposition was partly inspired by important books in the field such as those by
Bennett and Sharpley [14], Maz’ya [149], Krasnosel’skii and Rutitskii [123], Rana
[185], Diening, Hästö, Harjulehto and R˚užiˇcka [63] and others.
Luboš Pick wishes to express his special deep gratitude to Ms. Lenka Slavíková for
many stimulating discussions and suggestions that led to great improvement of some
parts of the text.
We thank Mr. Komil Kuliev, Ms. Guli Kulieva and Ms. Eva Ritterová for their help
with the preparation of the manuscript in L
ATEX.
We would like to thank Ms. Anja Möbius from the publishing house De Gruyter
for her collaboration and mostly for her patience.

13. Preface ix
Finally, the authors would be grateful for critical comments and suggestions for later
improvements.
The three authors below would like to dedicate this second edition to the memory of
the fourth author, the late Professor Svatopluk Fuˇcík, who passed away prematurely
not long after the appearance of the first edition of the book [126].
Prague, September 2012 Luboš Pick, Alois Kufner and Old ˇrich John

15. Contents
Preface v
1 Preliminaries 1
1.1 Vectorspace.............................................. 1
1.2 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Metric,metricspace ....................................... 6
1.4 Norm,normedlinearspace .................................. 6
1.5 Modular spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Inner product, inner product space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Convergence,Cauchysequences.............................. 11
1.8 Density, separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.9 Completeness............................................. 12
1.10 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.11 Products of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.12Schauderbases............................................ 14
1.13Compactness ............................................. 15
1.14Operators(mappings) ...................................... 16
1.15Isomorphism,embeddings................................... 18
1.16 Continuous linear functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.17 Dual space, weak convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.18 The principle of uniform boundedness . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.19Reflexivity ............................................... 21
1.20 Measure spaces: general extension theory . . . . . . . . . . . . . . . . . . . . . . 22
1.21TheLebesguemeasureandintegral ........................... 29
1.22Modesofconvergence...................................... 34
1.23 Systems of seminorms, Hahn–Saks theorem . . . . . . . . . . . . . . . . . . . . . 36

16. xii Contents
2 Spaces of smooth functions 38
2.1 Multiindices and derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Classes of continuous and smooth functions . . . . . . . . . . . . . . . . . . . . . 39
2.3 Completeness............................................. 43
2.4 Separability, bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5 Compactness ............................................. 51
2.6 Continuous linear functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7 Extensionoffunctions...................................... 59
3 Lebesgue spaces 62
3.1L
p
-classes............................................... 62
3.2 Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 Meancontinuity........................................... 67
3.4 Mollifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Densityofsmoothfunctions ................................. 71
3.6 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.7 Completeness............................................. 72
3.8 Thedualspace............................................ 74
3.9 Reflexivity ............................................... 78
3.10 The spaceL
1
............................................ 78
3.11 Hardy inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.12 Sequence spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.13Modesofconvergence...................................... 93
3.14Compactsubsets .......................................... 94
3.15Weakconvergence ......................................... 95
3.16 Isomorphism ofL
p
./andL
p
.0; .//...................... 96
3.17Schauderbases............................................ 97
3.18 Weak Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.19Remarks................................................. 104
4 Orlicz spaces 108
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2 Young function, Jensen inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3 Complementaryfunctions ................................... 115

17. Contents xiii
4.4 The 2-condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.5 ComparisonofOrliczclasses ................................ 122
4.6 Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.7 Hölder inequality in Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.8 TheLuxemburgnorm ...................................... 134
4.9 Completeness of Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.10 Convergence in Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.11 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.12 The spaceE
ˆ
./......................................... 145
4.13 Continuous linear functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.14 Compact subsets of Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.15 Further properties of Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4.16Isomorphismproperties,Schauderbases........................ 163
4.17 Comparison of Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5 Morrey and Campanato spaces 173
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.2 Marcinkiewicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.3 Morrey and Campanato spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.4 Completeness............................................. 178
5.5 Relations to Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.6 Somelemmas............................................. 181
5.7 Embeddings .............................................. 185
5.8 The John–Nirenberg space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.9 Another definition of the spaceJN.Q/........................ 194
5.10 SpacesN
p;.Q/........................................... 197
5.11Miscellaneousremarks ..................................... 199
6 Banach function spaces 203
6.1 Banach function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.2 Associatespace ........................................... 209
6.3 Absolutecontinuityofthenorm .............................. 216
6.4 Reflexivity of Banach function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 223
6.5 Separability in Banach function spaces . . . . . . . . . . . . . . . . . . . . . . . . . 228

18. xiv Contents
7 Rearrangement-invariant spaces 237
7.1 Nonincreasingrearrangements ............................... 237
7.2 Hardy–Littlewood inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
7.3 Resonant measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.4 Maximal nonincreasing rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . 249
7.5 Hardylemma ............................................. 251
7.6 Rearrangement-invariant spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
7.7 Hardy–Littlewood–Pólya principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
7.8 Luxemburgrepresentationtheorem............................ 256
7.9 Fundamental function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
7.10 Endpoint spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
7.11Almost-compactembeddings ................................ 275
7.12Gouldspace .............................................. 292
8 Lorentz spaces 301
8.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
8.2 Embeddings between Lorentz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 305
8.3 Theassociatespace ........................................ 307
8.4 The fundamental function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
8.5 Absolutecontinuityofnorm ................................. 309
8.6 Remarks onkk
1;1....................................... 311
9 Generalized Lorentz–Zygmund spaces 313
9.1 Measure-preservingtransformations ........................... 313
9.2 Basicproperties ........................................... 314
9.3 Nontriviality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
9.4 Fundamental function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
9.5 Embeddings between Generalized Lorentz–Zygmund spaces . . . . . . . 320
9.6 Theassociatespace ........................................ 332
9.7 When Generalized Lorentz–Zygmund space is Banach function space 353
9.8 Generalized Lorentz–Zygmund spaces and Orlicz spaces . . . . . . . . . . 356
9.9 Absolutecontinuityofnorm ................................. 367
9.10 Lorentz–Zygmund spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
9.11 Lorentz–Karamata spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

19. Contents xv
10 Classical Lorentz spaces 375
10.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
10.2Functionalproperties ....................................... 380
10.3Embeddings .............................................. 388
10.3.1 Embeddings of typeƒ,!ƒ.......................... 392
10.3.2 Embeddings of typeƒ,!........................... 393
10.3.3 Embeddings of type,!ƒ........................... 396
10.3.4 Embeddings of type,!........................... 399
10.3.5 TheHalperinlevelfunction............................ 401
10.3.6 Embeddings of type
p;1
.v/ ,!ƒ
q
.w/................. 404
10.3.7 The single-weight case
1;1
.v/ ,!ƒ
1
.v/............... 406
10.4 Associate spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
10.5 Lorentz and Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
10.6 Spaces measuring oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
10.7Themissingcase .......................................... 425
10.8Embeddings .............................................. 427
10.8.1 Embeddings of typeS,!S........................... 429
10.8.2 Embeddings of type,!SandS,!................. 431
10.8.3 Embeddings of typeƒ,!SandS,!ƒ................ 434
11 Variable-exponent Lebesgue spaces 437
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
11.2Basicproperties ........................................... 438
11.3Embeddingrelations ....................................... 445
11.4Densityofsmoothfunctions ................................. 447
11.5Reflexivityanduniformconvexity ............................ 450
11.6 Radon–Nikodým property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
11.7Daugavetproperty ......................................... 455
Bibliography 459
Index 472

21. Chapter 1
Preliminaries
In this chapter we give a survey of concepts and results from functional analysis that
will be used in the text. All results are stated without proofs, which can be found in
standard monographs.
1.1 Vector space
LetXbe a set of elements denoted byu;v;w;:::.
Definition 1.1.1.LetadditioninXbe defined, i.e. to every pairu2X,v2Xthere
corresponds an elementw2Xcalled thesumofuandvand denoted byuCv:
wDuCv:
Definition 1.1.2.Letmultiplicationby scalars be defined inX, i.e. to every real
number(called ascalar)andeveryu2Xthere corresponds an elementw2X
called the-multipleofuand denoted byu(oru):
wDu:
Definition 1.1.3.The setXwith addition and multiplication by scalars defined in it
is called areal vector spaceif the following axioms are satisfied:
(i)uCvDvCu (symmetry);
(ii)uC.vCw/D.uCv/Cw
(associativity);
(iii) inXthere exists a uniquely determined element denoted byand called the
zero elementsuch that
uCDu
for everyu2X;
(iv) for eachu2Xthere exists a uniquely determined element inXdenoted byu
such that
uC.u/DI
(v).uCv/DuCv,2R;
(vi).C/uDuCu,;2R;

22. 2 Chapter 1 Preliminaries
(vii).u/D./u,;2R;
(viii)1uDu;
(ix)0uD.
Definition 1.1.4.In a vector spaceX,thedifference(orsubtraction)uvof two
elementsu; v2Xis defined by
uvWDuC.v/:
Definition 1.1.5.LetMbe a subset of a vector spaceX. Denote
ŒM WD
\
Y;
where the intersection is taken over all vector spacesYXcontainingM.Then
ŒM is called thelinear hullofM. Obviously,ŒM is again a vector space.
Example 1.1.6.In what follows, the elements of a vector spaceXwill usually be
real-valuedfunctionsdefined on a certain setR:
uDu.x/; x2R:
In this case addition and scalar multiplication are defined as usual:
.uCv/.x/WDu.x/Cv.x/; .u/.x/WDu.x/:
Remark 1.1.7.We shall speak frequently of alinear spaceorlinear setinstead of a
vector space
.
Definition 1.1.8.A vector spaceXis called analgebraif for every ordered pair
u2X,v2X,aproductuvis defined as an element ofX, which satisfies the
following axioms for everyw2Xand all scalarsand:
(i).uv/wDu.vw/;
(ii)u.vCw/DuvCuw;
(iii).uCv/wDuwCvw;
(iv).u/.v/D./.uv/.
1.2 Topological spaces
Notation 1.2.1.LetXbe a nonempty set. Then by expXwe denote the set of all
subsets of setX.IfAX, we denote byA
c
the complement ofAwith respect toX,
that is,XnA.

23. Section 1.2 Topological spaces 3
Definition 1.2.2.We say that a couple.X;T/is atopological spaceifXis a nonempty
set andTis a system of subsets ofXsatisfying the following three conditions:
(i);2TandX2T(where;denotes the empty set).
(ii) IfG
12TandG 22T,thenG 1\G22T.
(iii) IfAis an index set of arbitrary cardinality andA
˛2Tfor every˛2A,then
S
˛2A
A˛2T.
The subsets ofXbelonging toTare calledopen setsin the spaceX,andthe
familyTis called atopologyonX.
Definition 1.2.3.Let.X;T/be a topological space. Ifx2X,B2Tandx2B,
then we say thatBis aneighborhoodofx.
Definition 1.2.4.Let.X;T/be a topological space. A familyBexpXis called
abasefor topologyTonXif every nonempty open subset ofXcan be represented
as a union over sets fromB.
Remark 1.2.5.Let.X;T/be a topological space. Then every baseBof the topol-
ogyThas the following properties:
(i) For everyG
1;G22Band every pointx2G 1\G2there exists a setG2B
such thatx2GG
1\G2.
(ii) For everyx2Xthere exists a setG2Bsuch thatx2G.
Proposition 1.2.6.LetXbe a nonempty set. LetBbe a collection of subsets ofX,
which has properties(i)and(ii)of Remark 1.2.5. We denote byTthe collection of all
those subsets ofX, which can be represented as unions of sets from some subcollection
ofB.ThenTis a topology onXandBis a base for this topology.
Definition 1.2.7.LetX,BandTbe as in Proposition 1.2.6. Then we say that the
topologyTisgenerated by the baseB.
Definition 1.2.8.Let.X;T/be a topological space and letF2expX. We say thatF
isclosedifF
c
2T.
Remark 1.2.9.Let.X;T/be a topological space. Denote byFthe system of all
closed subsets ofX. Then it follows from Definition 1.2.2 and De Morgan laws that
(i);2FandX2F.
(ii) IfF
12FandF 22F,thenF 1[F22F.
(iii) IfAis an index set of arbitrary cardinality andF
˛2Ffor every˛2A,then
T
˛2A
F˛2F.

24. 4 Chapter 1 Preliminaries
Definition 1.2.10.Let.X;T/be a topological space and letAX. Then the set
AWD
\
¹F2FIFAº
is called theclosureofAinX(with respect to the topologyT).
Theorem 1.2.11.Let.X;T/be a topological space. The closure operator has the
following properties:
(i);D;
(ii)AA
(iii)A[BDA[B
(iv).A/DA
Proposition 1.2.12.LetXbe a nonempty set. Assume thatclWexpX!expXis
an operator assigning to every setA2expXsome setcl.A/DA2expXsuch that
the properties(i)–(iv)of Theorem 1.2.11 hold. Then the family
TWD¹XnAIADcl.A/º (1.2.1)
is a topology onX. Moreover, the setcl.A/DAis the closure ofAinXwith respect
toT.
Definition 1.2.13.LetXbe a nonempty set and letT
1,T2be two topologies onX.
We say that the topologyT
1isweakerthanT 2(that is,T 2isstrongerthanT 1)if
T
1T2.
Definition 1.2.14.Let.X;T/and.Y;T
0
/be two topological spaces. A mapping
fWX!Yis calledcontinuousiff
1
.G/2Tfor everyG2T
0
.
Example 1.2.15.LetXDR, the set of all real numbers. Then the system
TWD
´
1
[
nD1
.an;bn/
μ
;
where.a
n;bn/are pairwise disjoint nontrivial open intervals, complemented with;
andRitself, defines a natural topology onR.
Notation 1.2.16.Let.X;T/and.Y;T
0
/be two topological spaces. Then byC.X;Y /
we denote the set of all continuous mappings fromXtoY.

25. Section 1.2 Topological spaces 5
Definition 1.2.17.Let.X;T/be a topological space and let¹f nº
1
nD1
be a sequence
of functions fromXtoR.LetfWX!Rbe a function. We say that¹f
nº
1
nD1
isuniformly convergenttofif, for every">0, there exists ann 02Nsuch that for
everyn2N,nn
0and for everyx2X, one hasjf n.x/f.x/j<". We write
fDlim
n!1
fn:
Our next aim is to develop some natural topologies onC.X;Y /. We begin with the
special case whenYDR.
Definition 1.2.18.Let.X;T/be a topological space. We thendefine the operator cl
on expC.X;R/in the following way: given a setAC.X;R/,weset
cl.A/WD
°
f2XIfDlim
n!1
fnfor some sequence¹f nº
1
nD1
A
±
:(1.2.2)
Proposition 1.2.19.Let.X;T/be a topological space and letclbe the operator
onexpC.X;R/defined by(1.2.2).Thenclsatisfies the properties(i)–(iv)of Theo−
rem 1.2.11.
Definition 1.2.20.Let.X;T/be a topological space and let cl be the operator on
expC.X;R/defined by (1.2.2). Then the topology generated by this closure operator
onC.X;R/through the formula (1.2.1), is called thetopology of uniform conver−
gence.
Now we shall define a reasonable topology on the setC.X;Y /for an arbitrary pair
of topological spaces.
Definition 1.2.21.LetXandYbe topological spaces. For every pair of setsA2
expXandB2expY, we denote
M.A;B/WD¹f2C.X;Y /If.A/Bº:
LetT
Ybe the topology ofY. Further, denote byF Xthe family of all finite subsets
ofX. Define next the system
BWD
8
<
:
k
\
iD1
M.Ai;Gi/; Ai2FX;Gi2T;iD1;:::;k
9
=
;
:
Then the setB(according to Proposition 1.2.6) generates a topology onC.X;Y /.
This topology is called thetopology of pointwise convergence.
Remark 1.2.22.For every topological spaceX, the topology of uniform convergence
is stronger onC.X;R/than the topology of pointwise convergence.
Proof of the assertions in this section as well as many further details on general
topological spacescan be found, e.g., in [71].

26. 6 Chapter 1 Preliminaries
1.3 Metric, metric space
Definition 1.3.1.LetXbe a nonempty set. A nonnegative function%defined on the
Cartesian productXXis called ametricif it satisfies the following axioms for every
u; v; w2X:
(i)%.u; v/D0if and only ifuDv;
(ii)%.u; v/D%.v; u/(symmetry);
(iii)%.u; v/5%.u; w/C%.w; v/(the triangle inequality).
AsetXwith a metric%will be called ametric spaceand denoted by.X; %/.
Definition 1.3.2.Let.X; %/be a metric space and letGX. We say thatGisopen
inXwith respect to%if for everyx2Gthere exists anr>0such that the set
B
r.x/WD ¹y2XI%.x; y/ < rº
satisfiesB
r.x/G.ThesetB r.x/is called anopen ball centered atxwith radiusr.
Remark 1.3.3.If.X; %/is a metric space, then the metric%automatically generates
a topology onX, in which open sets are those that are open with respect to%.Itisan
easy exercise to verify that the axioms of Definition 1.2.2 are satisfied.
1.4 Norm, normed linear space
Definition 1.4.1.LetXbe a vector space. Anonnegative function defined onX
whose value atu2Xis denoted bykukis called anormonXif it satisfies the
following axioms:
(i)kukD0if and only ifuD;
(ii)kukDjjkukfor everyu2Xand all scalars(the homogeneity axiom);
(iii)kuCvk5kukCkvkfor everyu; v2X(the triangle inequality).
A vector spaceXendowed with a normkkis called anormed linear space;the
numberkukis called the norm ofu2X.
Remark 1.4.2.If it is necessary to specify the vector spaceXon which the norm is
defined we usekuk
Xinstead ofkuk. Sometimes the normed linear spaceXwill be
denoted by.X;kk
X/.
Definition 1.4.3.If the functionkksatisfies only the axioms (ii), (iii) from Defini-
tion 1.4.1 then it is called aseminorm(or sometimes apseudonorm).

27. Section 1.5 Modular spaces 7
If the functionkksatisfies the axioms (i), (ii) from Definition 1.4.1 and there exists
a constantC>1such that (iii) becomes
(iii
0
)kuCvk5C.kukCkvk/for everyu; v2X,
then it is called aquasinorm.
Proposition 1.4.4.Every normed linear space is a metric space with the metric%
defined by
%.u; v/WD kuvk;u;v2X:
Hence, owing to Remark 1.3.3, it is also a topological space.
Definition 1.4.5.LetXbe a normed linear space,u
02X,r>0.Theset
B.u
0;r/WD ¹u2XIkuu 0k<rº
is called anopen ball(with centeru
0and radiusr).
A subsetMXis called anopen setinXif for everyu
02Mthere exists an
rDr.u
0/>0such thatB.u 0;r/M.
A subsetMXis called aclosed setinXifXnMis an open set inX.
1.5 Modular spaces
Definition 1.5.1.LetXbe a vector space. A function%WX!Œ0;1is called
left−continuousif the mapping7 !%.x/is continuous onŒ0;1/in the sense that
lim
!1
%.x/D%.x/for everyx2X:
A convex and left-continuous function%WX!Œ0;1is called asemimodular
onXif
(i)%./D0;
(ii)%.x/D%.x/for everyx2X;
(iii) if%.x/D0for every2R,thenxD.
A semimodular is called amodularonXif%.x/D0if and only ifxD.
A semimodular is calledcontinuousif the mapping7 !%.x/is continuous on
Œ0;1/for every fixedx2X.
Definition 1.5.2.LetXbe a vector space and let%be a semimodular or a modular
onX. Then the space
X
%WD
²
x2XIlim
!0
%.x/D0
³
is called asemimodular spaceor amodular space, respectively.

28. 8 Chapter 1 Preliminaries
Notation 1.5.3.LetXbe a vector space and let%be a semimodular onX.Wethen
denote bykk
%the functional, given for everyx2Xby
kxk
%WDinf
²
>0I%

1

x

1
³
: (1.5.1)
Remark 1.5.4.LetXbe a vector space and let%be a semimodular onX.Thenit
can be easily shown that the functionalkk
%defined by (1.5.1) is a norm onX %.It
is also known as theMinkowski functionalof the set¹x2XI%.x/1º. A detailed
proof can be found for example in [63, Theorem 2.1.7].
Definition 1.5.5.LetXbe a vector space and let%be a semimodular onX. Then the
functionalkk
%defined by (1.5.1) is called theLuxemburg normonX %.
Remark 1.5.6.LetXbe a vector space and let%be a semimodular onX. Then, for
every fixedx2X, the mapping7 !%.x/is nondecreasing onŒ0;1/. Moreover,
by the convexity of%,wehave
%.x/
´
%.x/for every2?0; 1;
%.x/for every2Œ1;1/:
(1.5.2)
Proposition 1.5.7.LetXbe a vector space, let%be a semimodular onXand let
x2X.Then
%.x/1if and only ifkxk
%1:
Proof.Assume that%.x/1. Then the definition of the Luxemburg norm also
implies thatkxk
%1.
Now assume thatkxk
%1. Then, for every>1, one has%.
1
x/1.Since%
is left-continuous, we obtain%.x/1. The proof is complete.
Proposition 1.5.8.LetXbe a vector space, let%be a semimodular onXand let
x2X.
(i)Ifkxk
%1,then%.x/kxk %.
(ii)Ifkxk
%>1,then%.x/kxk %.
(iii)For everyx2X,kXk
%%.x/C1.
Proof.(i) WhenxD0, there is nothing to prove. Assume that0<kxk
%1.By
Proposition 1.5.7 and by the fact thatkxkxk
1
%
k%D1, we obtain
%

x
kxk%

1:
Becausekxk
%1, the claim follows from (1.5.2).

29. Section 1.5 Modular spaces 9
(ii) Ifkxk %>1, then for every2.1;kxk %/,wehave%.
1
x/ > 1. Thus,
by (1.5.2), we obtain
1
%.x/ > 1. Since this is so for an arbitrary<kxk %,
the claim follows.
(iii) This is an immediate consequence of (ii).
Proposition 1.5.9.LetXbe a vector space, let%be a semimodular onXand let
¹x
nº
1
nD1
be a sequence inX.Thenlim n!1kxnk%D0if and only iflim n!1%.xn/
D0for every>0.
Proof.Assume first that lim
n!1kxnk%D0and let>0. Then, for everyK>1,
there exists ann
02Nsuch that for everyn2N,nn 0, one has
kKx
nk%<1:
Therefore also%.Kx
n/1and, by (1.5.2),
%.x
n/D%

1
K
Kx
n


1
K
%.Kx
n/
1
K
;
establishing lim
n!1%.xn/D0.
Now assume that lim
n!1%.xn/D0. Then there exists ann 02Nsuch that for
everyn2N,nn
0, one has%.x n/1. Especially, for suchn,wehave
kx
nk%
1

:
Sincewas arbitrary, we obtain lim
n!1kxnk%D0, as desired. The proof is com-
plete.
Definition 1.5.10.LetXbe a vector space, let%be a modular onXand let¹x nº
1
nD1
be a sequence inX. We say that¹x nº
1
nD1
ismodular convergentto somex2Xif
lim
n!1
%.xnx/D0:
Corollary 1.5.11.It follows from Proposition 1.5.8 that ifXis a modular space, then
the norm convergence always implies the modular convergence.
Proposition 1.5.12.LetXbe a modular space. Then the modular convergence onX
is equivalent to the norm convergence if and only if%.x
n/!0implies%.2x n/!0.
Proof.)Assume that the modular convergence onXis equivalent to the norm.
Let¹x
nº
1
nD1
be a sequence inXsuch that%.x n/!0.Also,kx nk%!0,andit
follows immediately from Proposition 1.5.8 applied toD2that%.2x
n/!0.
(Assume conversely that the condition holds. Let¹x
nº
1
nD1
be a sequence inX
satisfying%.x
n/!0.Given>0,wefindm2Nsuch that2
m
. Then,

30. 10 Chapter 1 Preliminaries
iterating the conditionmtimes, we obtain%.2
m
xn/!0asn!1. Thus, by (1.5.2),
we obtain
0lim
n!1
%.xn/2
m
lim
n!1
%.2
m
xn/D0:
Owing to Proposition 1.5.9, this establishesx
n!0, as desired. The proof is com-
plete.
Theorem 1.5.13.Let.X; % X/and.Y; % Y/be two modular spaces,XY. Assume
that there exists a functionhW.0;1/!.0;1/, which is bounded on some right
neighborhood of zero. Suppose that
%
Y.x/h.% X.x//for everyx2X: (1.5.3)
Then there exists a positive constantCsuch that, for everyx2X,
kxk
%Y
Ckxk %X
: (1.5.4)
Proof.Suppose that the assertion is not true. Then for everyn2Nthere exists some
x
n2Xsuch thatkx nk%Y
>nkx nk%X
.Set
Qx
nWD
x
n
kxnk
p
n%X
;n2N:
ThenkQx
nk%X
!0andkQx nk%Y
!1. By Proposition 1.5.8, we get% X.Qxn/!0and
%
Y.Qxn/!1. That, however, is a contradiction with (1.5.3), sincehis bounded on
some neighborhood of zero. The proof is complete.
For more details on modularspaces, see, e.g., [63, 160, 163]. Most of the material
in this section can be found in [63]. Theorem 1.5.13 is a special case of a more general
result in [125], see also [178].
1.6 Inner product, inner product space
Definition 1.6.1.LetXbe a vector space. A real-valued function onXX, whose
value at the ordered pair.u; v/,u; v2X, is denoted byhu; vi, is called aninner
productonXif it satisfies the following axioms:
(i)hu; ui>0for everyu2X,u6 D;
(ii)hu; viDhv;uifor everyu; v2X;
(iii)huCv;wiDhu; wiChv;wifor everyu; v; w2X;
(iv)hu; viDhu; vifor everyu; v2Xand all scalars.
A vector spaceXendowed with an inner product is called aninner product space
(or aunitary space).

31. Section 1.7 Convergence, Cauchy sequences 11
Definition 1.6.2.LetXbe a unitary space. Foru2X,wedefine
kukDhu; ui
1
2:
The inequalities
jhu; vij5kukkvk;u;v2X;
(the so-calledCauchy–Schwarz inequality)and
kuCvk5kukCkvk;u;v2X;
holds. In particular, every unitary space is a normed linear space with respect to the
normkukgenerated by the inner producthu; vi.
1.7 Convergence, Cauchy sequences
Definition 1.7.1.LetXbe a metric space with respect to the metric%and let¹u nº
1
nD1
be a sequence inX. We say thatu nconverges tou2X(and writeu n!uinX)if
lim
n!1%.un;u/D0,i.e.ifforevery">0there exists ann 0Dn0."/2Nsuch
that%.u
n;u/<"for alln>n 0.
We shall also say thatu
nconverges toustronglyinXorin the norm ofX.
Ifu
n!uinX, then the sequence¹u nº
1
nD1
is said to beconvergentinXanduis
called thelimitof¹u
nº
1
nD1
.
Definition 1.7.2.LetXbe a normed linear space, let¹u
nº
1
nD1
be a sequence inX
and letu2X.If
lim
n!1





u
n
X
kD1
u
k





X
D0;
we say that the series
P
1
nD1
unconverges touinXand write
uD
1
X
nD1
un:
Definition 1.7.3.A sequence¹u
nº
1
nD1
in a metric space.X; %/is called aCauchy
sequenceif
lim
m;n!1
%.um;un/D0;
i.e. if for every">0there exists ann
0Dn0."/2Nsuch that%.u m;un/<"for all
m; n2N,m; n > n
0.
Remark 1.7.4.In every metric space, each convergent sequence is a Cauchy se-
quence. The converse is not true in general.

32. 12 Chapter 1 Preliminaries
The following simple auxiliary assertion will be useful later.
Lemma 1.7.5.Let¹u
nº
1
nD1
be a Cauchy sequence in a metric spaceXand let
¹u
nk
º
1
kD1
be its subsequence. Ifu nk
!uinXthenu n!uinX.
Proof.Let">0. Then, by the Cauchy property, there exists an indexn
02Nsuch
that for everym; n2N,m; nn
0, one has%.x m;xn/<". Next, there is an index
k
02Nsuch thatn
k0
n0and%.x nk
0
;x/ <". Thus, for everyn2N,nn
k0
, one
has
%.x
n;x/%.x n;xnk
0
/C%.x nk
0
;x/ <"C"D2":
The proof is complete.
Definition 1.7.6.LetMbe a subset of a metric spaceX.TheclosureofMinX,
denoted byMorM
X
, is defined as the set of all elementsu2Xsuch that there
exists a sequence¹u
nº
1
nD1
inMfor whichu n!uinX.
Remark 1.7.7.Clearly, ifMis a subset of a metric spaceX,thenM
M.AsetM
is closed inXif and only ifMDM.
1.8 Density, separability
Definition 1.8.1.A subsetMof a metric spaceXis said to bedenseinXifMDX.
Definition 1.8.2.A metric spaceXis calledseparableif it contains a countable dense
subset.
Remark 1.8.3.Let.X; %/be a metric space. Assume that there exists an uncountable
subsetMofXand aı>0such that
%.u; v/ > ıfor everyu; v2M; u6 Dv:
ThenXis not separable.
1.9 Completeness
Definition 1.9.1.A metric spaceXis said to becompleteif every Cauchy sequence
inXis convergent inX.
Definition 1.9.2.A complete normed linear space is called aBanach space.
Definition 1.9.3.LetXbe a unitary space endowed with an inner producthu; vi.
If the normed linear spaceXis complete with respect to the normkukDhu; ui
1
2,
thenXis called aHilbert space.

33. Section 1.10 Subspaces 13
Definition 1.9.4.We say that a normed linear space.X;kk X/has theRiesz–Fischer
propertyif for each sequence¹u
nº
1
nD1
such that
1
X
nD1
kunkX<1; (1.9.1)
there exists an elementu2Xsuch that
P
1
nD1
unDuinX,thatis,
lim
n!1





n
X
kD1
u
ku





X
D0:
Theorem 1.9.5.A normed linear space is complete if and only if it has the Riesz–
Fischer property.
Proof.Assume first thatXis a Banach space and let¹u
nº
1
nD1
be a sequence inX
such that (1.9.1) holds. Then
´
n
X
kD1
u
k
μ
1
nD1
is a Cauchy sequence inX, hence it converges inXto some elementu2X. Thus,
uD
1
X
nD1
un:
Conversely, assumeXis a normed linear space with the Riesz–Fischer property
and let¹u
nº
1
nD1
be a Cauchy sequence inX. Then a subsequence¹u nk
º
k2Ncan be
chosen so that
1
X
kD1
kunk
unkC1
kX<1:
Then, necessarily, the series
P
1
kD1

u
nk
unkC1

converges inX. In particular,
there exists an elementu2Xsuch thatu
nk
!uinX. But then since¹u nº
1
nD1
is
a Cauchy sequence, we also haveu
n!uinX, as desired.
1.10 Subspaces
Definition 1.10.1.A subset of a normed linear spaceXis called asubspaceofXif
it is a linear set which is closed inX.
Remark 1.10.2.We shall distinguish betweenlinear subsetsandsubspaces: a linear
subsetneed not be closedinX.

34. 14 Chapter 1 Preliminaries
Remarks 1.10.3.(i) A subspaceMof a normed linear space.X;kk X/is again
a normed linear space with the normkk
Mdefined by
kuk
MWD kuk Xforu2M:
(ii) A subspace of a separable normed linear spaceXis itself a separable normed
linear space.
(iii) A subspace of a Banach space is also a Banach space.
1.11 Products of spaces
Remarks 1.11.1.(i) Letn2N,andletX 1;X2;:::;Xnbe normed linear spaces,
XDX
1X2X nthe Cartesian product ofX 1;:::;Xn,i.e.thesetof
all (ordered)n-tuples
uD.u
1;:::;un/
such thatu
i2Xi,iD1;:::;n.ThenXis also a normed linear space; the
norm inXcan be defined in various ways, for example
kuk
XWD

n
X
iD1
kuik
p
X
i
!1
p
for somep2Œ1;1/
or
kuk
XWDmax
iD1;:::;n
kuikXi
:
(ii) LetX
1;:::;Xnbe separable normed linearspaces. Then the product space
XDX
1X nis also a separable normed linear space.
(iii) LetX
1;:::;Xnbe Banach spaces. Then the product spaceXDX 1X n
is also a Banach space.
1.12 Schauder bases
Definition 1.12.1.LetXbe a Banach space. A sequence¹u nº
1
nD1
inXis called a
Schauder basisofXif for everyu2Xthere exists a unique sequence¹a
nº
1
nD1
of
scalars such that
uD
1
X
nD1
anun:
The (uniquely determined) numbersa
nDan.u/are called thecoefficientsofuwith
respect to the Schauder basis¹u
nº
1
nD1
.

35. Section 1.13 Compactness 15
Remark 1.12.2.Every Banach space with a Schauder basis is separable. The con-
verse implication does not hold, i.e. there exists a separable Banach space without
Schauder basis.
Definition 1.12.3.A Schauder basis¹u
nº
1
nD1
in a Banach spaceXis calleduncon−
ditionalif the convergence of a series of the form
1
X
nD1
anun
implies the convergence of the series
1
X
nD1
a
.n/u
.n/
for every permutation of the setN.
Remark 1.12.4.Abasis¹u
nº
1
nD1
is unconditional if and only if at least one of the
following conditions holds:
(i) The convergence of a series
P
1
nD1
anunimplies the convergence of the series
P
1
nD1
"nanunfor any choice of" nequal toC1or1.
(ii) The convergence of a series
P
1
nD1
anunimplies the convergence of the series
P
1
kD1
ank
unk
for any subsequence¹n
kº
1
kD1
ofN.
1.13 Compactness
Definition 1.13.1.Let.X; %/be a metric space. A setMXis said to berel−
atively compactif every sequence inMcontains a convergent subsequence (i.e. a
subsequence with a limit inX).
IfMis closed and relatively compact thenMis said to becompact.
Definition 1.13.2.Let.X; %/be a metric space. Let">0and letMbe a subset
ofX.AsetEinXis called an"-netofMif for everyu2Mthere exists au
"2E
such that
%.u; u
"/<":
AsetMXis calledtotally bounded(orprecompact)inXif for every">0there
exists a finite"-net ofMinX.
Remarks 1.13.3.Let.X; %/be a metric space. Then
(i) every relatively compact set inXis bounded;

36. 16 Chapter 1 Preliminaries
(ii) every closed subset of a compact set is itself compact;
(iii) ifXis complete, then each its subset is relatively compact if and only if it is
totally bounded.
Remark 1.13.4.ForiD1;:::;n,letM
ibe a relatively compact subset of a Banach
spaceX
i.ThenM 1M nis a relatively compact subset ofX 1X n.
1.14 Operators (mappings)
Definition 1.14.1.LetX; Ybe normed linear spaces, andMasetinX. Suppose that
aruleisgivenbywhichtoeveryu2Mthere corresponds a uniquely determined
element inY. We denote this element by
AuorA.u/
and say that the rule defines anoperatorAonM.ThesetMis called thedomainof
the operatorAand denoted by Dom.A/;theset
Rng.A/D¹Qu2YIQuDAufor someu2Mº
is called therangeof the operatorA.
If for allu; v2Mwe haveAu6 DAvprovidedu6 Dv,thentoeveryQu2Rng.A/
is assigned a uniquely determined elementu2Mby the ruleAuDQu. We write this
as
uDA
1
Qu
and callA
1
theinverse operatortoA.WehavethatDom.A
1
/DRng.A/,Rng.A
1
/
DDom.A/DM.
We say that the operatorAis an operatorfromXintoYwhich mapsMintoY.If
Rng.A/DY, we say thatAmapsMontoY.
The expressionsfunction,abstract functionormappingare frequently used instead
ofoperator.
Definition 1.14.2.LetX; Ybe normed linear spaces. An operatorAfromXintoYis
calledlinearif Dom.A/is a linear set and if the following two conditions are satisfied:
(i)A.uCv/DAuCAv
(ii)A.u/DAu
for allu; v2Dom.A/and for every scalar.
Definition 1.14.3.LetX; Ybe normed linear spaces. An operatorAfromXintoY
is said to becontinuousif
u
n!uinXimpliesAu n!AuinY
for any sequence¹u
nº
1
nD1
providedu n2Dom.A/,u2Dom.A/.

37. Section 1.14 Operators (mappings) 17
Definition 1.14.4.LetX; Ybe normed linear spaces. A linear operatorAfromX
intoYis said to beboundedif
supkAuk
Y<1;
where the supremum is taken over allu2Dom.A/such thatkuk
X51.
Remark 1.14.5.A linear operator between twonormed linear spaces isbounded if
and only if it is continuous.
We shall now introduce the notion of anormof a continuous linear operator (the
so-calledoperator norm).
Definition 1.14.6.LetX; Ybe normed linear spaces. LetAbe a continuous linear
operator fromXintoY.Then
kAk
X!Y WDsup¹kAuk YIu2Dom.A/;kuk X51º:
Remark 1.14.7.LetX; Ybe normed linear spaces and letAbe a continuous linear
operator fromXintoY. Then the following formulas for the normkAk
X!Y ofA
will often be useful:
kAk
X!Y Dsup
u2Dom.A/
kuk
X
D1
kAuk YDsup
u2Dom.A/
u6 D
kAuk Y
kukX
:
Theorem 1.14.8(Banach).LetX; Ybe Banach spaces, andAa linear operator
fromXontoYwithDom.A/DX(andRng.A/DY). Suppose thatAis continuous
and thatA
1
exists. ThenA
1
is continuous.
Definition 1.14.9.LetAbe a linear operator from a normed linear spaceXinto a
normed linear spaceYwith Dom.A/DX. The operatorAis said to becompact(or
completely continuous) if it maps every bounded set inXonto a relatively compact
set inY.
Remark 1.14.10.Every compact linear operator is continuous.
Theorem 1.14.11(Banach–Steinhaus).LetXbe a Banach space,Ya normed lin−
ear space. The sequence¹A
nº
1
nD1
of bounded linear operators fromXintoYwith
Dom.A
n/DXsatisfies
A
nu!AuinYfor everyu2X
if and only if the following two conditions are satisfied:
(i)the sequence¹kA
nkº
1
nD1
is bounded;
(ii)A
nu!AuinYfor everyu2M,whereMis a dense subset ofX.

38. 18 Chapter 1 Preliminaries
1.15 Isomorphism, embeddings
Definition 1.15.1.Two normed linear spacesX; Yare said to beisomorphicif there
exists a continuous linear operatorAsuch that Dom.A/DX,Rng.A/DY,andA
1
exists and is continuous. This operatorAis called anisomorphism mappingor briefly
anisomorphismbetweenXandY.
Definition 1.15.2.(i) Two metric spaces.X; %/and.Y; /are said to beisometric
if there exists a mappingTsuch that Dom.T /DX,Rng.T /DYand
%.x; y/D.Tx;Ty/
for every pairx;y2X.
(ii) Two normed linear spacesX; Yare said to beisometrically isomorphicif there
exists a linear operatorAsuch that Dom.A/DX,Rng.A/DYand
kuvk
XDkAuAvk Y
for every pairu; v2X.
Remark 1.15.3.In particular, two isometrically isomorphic spaces are isomorphic.
Remark 1.15.4.LetX; Ybe two isomorphic normed linear spaces. Then
(i) ifXis separable thenYis also separable;
(ii) ifXis complete thenYis also complete;
(iii) ifXhas a Schauder basis thenYalso has a Schauder basis.
Definition 1.15.5.LetX; Ybe two normed linear spaces and letXY.Wedefine
theidentity operatorId fromXintoYwith Dom.Id/DRng.Id/DXas the operator
which maps every elementu2Xonto itself: IduDu, regarded as an element ofY.
If the identity operator is continuous, that is, if there exists a constantc>0such that
kuk
Y5ckuk Xfor everyu2X;
then we say that the spaceXisembedded into the spaceYand we shall call the
operator Id theembedding operatorfromXtoY. Alternatively, we may sometimes
say that there exists a continuous (or bounded) embedding ofXtoY.Bytheoperator
normof Id we call the number
kIdk
X,!Y WDsup
f6 0
kfkY
kfkX
:
We shall also call this quantity anembedding constant.

39. Section 1.16 Continuous linear functionals 19
Remarks 1.15.6.(i) The operator Id from Definition 1.15.5 is obviously linear.
(ii) While the elements inXandinRng.Id/Ycoincide, their respective norms
inXand inYmay be different.
Notation 1.15.7.IfXandYare two normed linear spaces and there exists a contin-
uous embedding fromXintoY, we write
X,!Y:
If simultaneously
X,!YandY,!X;
then we shall write
XY:
If the embedding operator is compact (see Definition 1.14.9), we write
X,!,!Y:
Definition 1.15.8.LetXbe a vector space and suppose thatkk
1andkk 2are two
norms onX. These norms are said to beequivalentif there exist constantsc>0,
d>0such that
ckuk
15kuk 25dkuk 1for allu2X:
In other words, the normskk
1andkk 2are equivalent if and only if.X;kk 1/
.X;kk
2/. In particular, the embedding from.X;kk 1/into.X;kk 2/is an isomor-
phism.
1.16 Continuous linear functionals
Definition 1.16.1.LetXbe a normed linear space. A linear operator fromXintoR
is then called alinear functional.
Remark 1.16.2.In this section we denote linear functionals by Greek letters:';ˆ;:::.
The value of the functional'atu2Xwill usually be denoted by
'.u/I
the notation
h';ui
is also very frequently used in the literature.
Since a functional is an operator, some concepts introduced in connection with
operators (see Section 1.14) can be transferred to functionals. We shall deal here only
withcontinuous linear functionals, that is, continuous linear operators fromXintoR.

40. 20 Chapter 1 Preliminaries
Definition 1.16.3.Anormof a linear functional'is defined by
k'kDsupj'.u/j
where the supremum is taken over allu2Dom.'/such thatkuk
X51.
Continuous linear functionals defined on linear subsets ofXcan beextendedto the
entire spaceXas stated in the following theorem.
Theorem 1.16.4(Hahn–Banach).Let'be a continuous linear functional defined on
a linear subsetMof a normed linear spaceX. Then there exists a continuous linear
functionalˆdefined onXsuch that
ˆ.u/D'.u/foru2MandkˆkDk'k:
1.17 Dual space, weak convergence
Definition 1.17.1.Let us denote byX

the set of all continuous linear functionals
defined onX.ThenX

is a vector space if we define the addition of functionals and
the multiplication of a functional by a scalar in the natural way, namely
.'C /.u/WD'.u/C .u/; .'/.u/WD'.u/
where'; 2X

andis a scalar.
Remarks 1.17.2.IfXis a normed linear space, then the spaceX

from Defini-
tion 1.17.1, endowed with the norm from Definition 1.16.3, isitself also a normed
linear space. Moreover, the spaceX

is a Banach space, that is, it is always complete.
Definition 1.17.3.LetXbe a normed linear space,¹u
nº
1
nD1
a sequence inX.Wesay
thatu
nconverges weaklytou2X(notationu n*uoru n
w!u)iflim n!1'.un/D
'.u/for every'2X

.
Remark 1.17.4.Every weakly convergent sequence is bounded.
Theorem 1.17.5(Banach–Steinhaus theorem for weak convergence).LetXbe a Ba−
nach space. The sequence¹u
nº
1
nD1
converges weakly tou2Xif and only if the
following two conditions are satisfied:
(i)the sequence¹ku
nkXº
1
nD1
is bounded;
(ii) lim
n!1'.un/D'.u/for all'2,whereis a dense subset of the dual
spaceX

.
Theorem 1.17.6(Banach–Alaoglu).LetXbe a Banach space andX

its dual. Then
the unit ball
®
ƒ2X

Ikƒk X
1
¯
is weakly* compact inX

.

41. Section 1.18 The principle of uniform boundedness 21
1.18 The principle of uniform boundedness
Theorem 1.18.1(Uniform boundedness principle).LetXbe a Banach space andY
a normed linear space. Let¹T
˛º˛2Ibe a set of linear operators fromXtoY,whereI
is an arbitrary index set (without any restriction on its cardinality). Assume that
sup
˛2I
kT˛xkY<1for everyx2X:
Then
sup
˛2I
kT˛kX!Y <1:
1.19 Reflexivity
Definition 1.19.1.LetXbe a Banach space,X

its dual space. Then we can define
the dual ofX

by setting
X

WD.X

/

:
Let us denote the elements ofX

byu

;v

;:::. The operatorJfromXinto
X

with Dom.J /DX, defined by the formula
.J u/.'/D'.u/for'2X

;u2X;
is called thecanonical mappingfromXintoX

.(Juis the elementu

2X

which satisfiesu

.'/D'.u/.)
Remark 1.19.2.LetXbe a Banach space. Denote byJ.X/the image ofXin the
canonical mappingJ.ThenJis an isometric isomorphism betweenXandJ.X/.
Definition 1.19.3.A Banach spaceXis said to bereflexiveif
J.X/DX

:
Remarks 1.19.4.(i) Every subspace of a reflexive Banach space is reflexive.
(ii) A Cartesian product of afinite number of reflexive Banach spaces is a reflexive
Banach space.
(iii) A Banach space isomorphic to a reflexive Banach space is reflexive.
(iv) If a Banach space has separable dual space, then it is itself separable.
(v) The dual space of a separable reflexive Banach space is separable.

42. 22 Chapter 1 Preliminaries
1.20 Measure spaces: general extension theory
Definition 1.20.1.LetXbe a nonempty set and letSexpXbe a collection of sub-
sets ofX.ThenSis called analgebraif the following three conditions are satisfied:
(i);;X2S;
(ii)A\B2Sfor everyA; B2S;
(iii)A
c
2Sfor everyA2S.
Remark 1.20.2.For every collectionFof subsets of a setX, there exists a unique
algebraSof subsets ofXsuch thatFSand ifS
1is another algebra containing
F,thenSS
1.
Definition 1.20.3.LetXbe a nonempty set and letSexpXbe a collection of
subsets ofX. Then every functionWS!Œ0;1is called aset functiononS.We
say that a set functionismonotoneonSif
.A/.B/wheneverA; B2S;AB:
We say thatisfinitely additiveonSif


n
[
iD1
Ai
!
D
n
X
iD1
.Ai/;
for everyn2Nand pairwise disjoint setsA
i2S,iD1;:::;n, such that
S
n
iD1
Ai2
S. We say thatiscountably additiveonSif


1
[
nD1
An
!
D
1
X
nD1
.An/
for all pairwise disjoint setsA
n2S,n2N, such that
S
1
nD1
An2S. We say that
iscountably subadditiveonSif
.A/
1
X
nD1
.An/
for everyA2Ssuch thatA
S
1
nD1
An,whereA n2Sfor eachn2N.
Definition 1.20.4.LetXbe a nonempty set and letSexpXbe a collection of
subsets ofXsatisfying;2S. A countably additive set functionWS!Œ0;1is
called ameasureonSif.;/D0.

43. Section 1.20 Measure spaces: general extension theory 23
We shall now extend a measure to a set function on the entire expX.Wepayforthis
extension by a possible loss of nice properties of the original measure. In particular,
the extended set function need not be countably additive any more.
Definition 1.20.5.LetXbe a nonempty set and letSexpXbe an algebra of
subsets ofXsatisfying;2S.LetWS!Œ0;1beameasureonS.Forevery
AXwe define


.A/WDinf
´
1
X
iD1
.Ai/IAi2S;A
1
[
iD1
Ai
μ
:
The function

is called theouter measure induced by.
Remark 1.20.6.An outer measure is well-defined since for everyA2expXthere
exists at least one sequence¹A
nº
1
nD1
such thatA
S
1
nD1
An. The set function

can attain infinite value.
In the next proposition we shall collect some properties of an outer measure.
Proposition 1.20.7.LetXbe a nonempty set and let

WexpX!Œ0;1be
an outer measure onX. Assume that

is induced by a measuredefined on an al−
gebraSexpX.Then
(i)

.A/0for everyA2expX;
(ii)

.;/D0;
(iii)

is monotone;
(iv)

is countably subadditive;
(v)

is an extension ofonSin the sense that

.A/D.A/for everyA2S.
We shall now extend the notion of an outer measure to set functions about which
we do notaprioriknow that they were induced by a measure.
Definition 1.20.8.LetXbe a nonempty set and letWexpX!Œ0;1beaset
function such that the properties (i)–(iv) of Proposition 1.20.7 are satisfied. Thenis
called anouter measureonX.
We have seen how an outer measure is defined on expXby an extension of a given
measure on a subalgebra of expX. Now we shall take the converse path. We start
with an outer measure and our aim will be to build a measure from it. This will be
done by choosing an appropriate subclass of expXon which the given outer measure
behaves like a measure.

44. 24 Chapter 1 Preliminaries
Definition 1.20.9.LetXbe a nonempty set and letWexpX!Œ0;1be an outer
measure onX. We say that a setA2expXis−measurableif
.T/D.T\A/C.T\A
c
/for every “test set”T2expX: (1.20.1)
We shall denote byMthe set of all-measurable subsets ofX.
Remark 1.20.10.IfA2M,thenalsoA
c
2Mdue to the symmetry in (1.20.1).
Definition 1.20.11.LetXbe a nonempty set and letSexpXbe a collection
of subsets ofX.ThenSis called a−algebraif the following three conditions are
satisfied:
(i);;X2S;
(ii)
S
1
nD1
An2Sfor every countable sequence¹A nº
1
nD1
S;
(iii)A
c
2Sfor everyA2S.
In the next proposition we shall collect some properties of the class of measurable
sets.
Proposition 1.20.12.LetXbe a nonempty set, letWexpX!Œ0;1be an outer
measure onX, and letMbe the set of all−measurable subsets ofX.Then
(i)Mis a−algebra of subsets ofX;
(ii)is countably additive when restricted toM;
(iii)ifwas induced by some measure defined on an algebraS,thenSM;
(iv)the set
NWD ¹A2expXI.E/D0º
satisfiesNM.
Remark 1.20.13.For every collectionAof subsets of a setX, there exists a unique-
algebraSof subsets ofXsuch thatASand ifS
1is another algebra containingA,
thenSS
1. In such cases, we can say that the-algebraSisgeneratedbyAand
writeSDS.A/.
Definition 1.20.14.LetXbe a topological space. LetGandFdenote the set of all
open and closed subsets ofX, respectively. Then
S.G/DS.F/:
We call the-algebra generated by open (or closed) sets the-algebra ofBorel subsets
ofXand denote it byB.X/.

45. Section 1.20 Measure spaces: general extension theory 25
Definition 1.20.15.LetXbe a nonempty set, letSexpXbe a collection of subsets
ofXand letWS!Œ0;1be a set function onS. We say thatisfiniteonSif
.A/ <1for everyA2S. We say thatis−finiteonSif there exists a sequence
¹X
nº
1
nD1
of pairwise disjoint subsets of setsX n2S,n2N, such that.X n/<1
for everyn2NandXD
S
1
nD1
Xn.
Definition 1.20.16.LetXbe a nonempty set, letSexpXbe a-algebra of subsets
ofXand letWS!Œ0;1beameasureonS. The pair.X;S/is then called
ameasurable spaceand the triple.X;S;/is called ameasure space. The elements
ofSare calledmeasurable sets.
Definition 1.20.17.Let.X;S;/be a measure space. Define
NWD¹A2expXIthere existsN2Ssuch thatANand.N /D0º:
Then the elements ofNare called the−null subsetsofX. We say that.X;S;/is
acompletemeasure space ifNS.
Remark 1.20.18.It is not difficult to realize that every measure can be completed.
Therefore it is not such a restriction to assume that the measure space in question is
complete.
Definition 1.20.19.Let.X;S;/be a measure space. A setA
2Sis called anatom
if.A/ > 0and for every setB2S,BA, either.B/D0or.AnB/D0.
The measure space.X;S;/is calledcompletely atomic(or justatomicordiscrete)
if there exists a setMXsuch that.XnM/D0and.¹xº/¤0for every
x2M. The measure space.X;S;/is callednonatomicif there do not exist any
atoms inS. We often for short say thatthe measureis nonatomic. Such measure is
also calledcontinuous.
Example 1.20.20.LetXbe a nonempty set and letSWDexpX.ForA2S,define
.A/WD
´
the number of elements ofAifAis finite;
1 ifAis infinite
:
Thenis a measure. This measure is called thecounting measureonX.
In the particular case whenXDN, the counting measure is denoted bymand is
called thearithmetic measureonN. The triple.N;expN;m/is a typical example of
a completely atomic space with all atoms having the same measure.
Definition 1.20.21.Let.X; %/be a metric space and letbe a measure defined on
B.X/. We say thatisouter regularif, for everyA2B.X/, one has
.A/Dinf¹.G/IGopen;AGº
Dsup¹.F /IFclosed;FAº:

46. 26 Chapter 1 Preliminaries
We say thatisinner regularif, for everyA2B.X/satisfying0<.A/<1and
every">0, there exists a compact setKAsuch that.AnK/ < ". If a measure
is both inner and outer regular, then we say that it isregular.
Definition 1.20.22.LetXbe a nonempty set andEX. The function
EWX!
Œ0;1/,definedby

E.x/WD
´
1ifx2E;
0ifx6 2E;
is called thecharacteristic functionofE.
Definition 1.20.23.Let.X;S;/be a complete-finite measure space. The function
sWX!Œ0;1/is called a−simple function(or justsimple function) if it is a finite
linear combination of characteristic functions of-measurable sets of finite measure,
i.e. if there exist anm2N, real numbers¹a
1;:::;amºand disjoint measurable
subsets ofXof finite measure¹E
1;:::;Emºsuch that
s.x/D
´
a
j;x2E j;jD1;:::;m;
0; x2Xn
S
m
jD1
Ej:
Definition 1.20.24.Let.X; %/be a metric space and letuWX!Œ0;1.Theset
¹x2XIu.x/6 D0º;
where the bar denotes the closure in the space.X; %/, is called thesupportof the
functionuand is denoted by suppu.
Remark 1.20.25.Let.X;S;/be a complete-finite measure space endowed fur-
ther with a metric. Then a functionsWX!Ris simple if and only if its range is
a finite set and its support is of finite measure.
Definition 1.20.26.Let.X;S;/be a complete-finite measure space and letA2
S. We say that a certain statement, sayV.x/, holdsalmost everywhereonA(ora.e.
onAfor short) with respect to(we will also sayfor almost allx2A)iftheset
EWD ¹x2AIV.x/does not holdº
satisfiesE2Sand.E/D0.
Definition 1.20.27.Let.X;S;/be a complete-finite measure space and letsbe
a simple function onXwith representation
s.x/D
m
X
iD1
aiEi
;x2X;

47. Section 1.20 Measure spaces: general extension theory 27
wherea i2RandE i2S,iD1;:::;m. We then define theintegralofsby
Z
X
s.x/d.x/WD
m
X
iD1
ai.Ei/:
Definition 1.20.28.Let.X;S;/be a complete-finite measure space and letfW
X!Œ0;1. We say thatfisS−measurable(or justmeasurable) if there exists
a nondecreasing sequence of nonnegative simple functions¹s
nº
1
nD1
satisfying
f.x/Dlim
n!1
sn.x/; x2X:
We shall denote byS
C.X/the set of all nonnegative measurable functions onX.
For a nonnegative measurable functionfWX!Œ0;1, we define itsintegralby
Z
X
f.x/d.x/WDlim
n!1
Z
X
sn.x/d.x/:
The first important property of nonnegative measurable functions is the following
theorem on monotone convergence.
Theorem 1.20.29(monotone convergence theorem).Let.X;S;/be a complete−
finite measure space and let¹f
nº
1
nD1
be a sequence of nonnegative integrable func−
tions defined onXsuch thatf
n.x/f nC1.x/for everyn2Nandx2X.Let
f.x/WDlim
n!1
fn.x/; x2X:
Thenf2S
C.X/and
Z

lim
n!1
fn.x/d.x/D
Z

f.x/d.x/Dlim
n!1
Z

fn.x/d.x/:
Our aim is now to extend the notion of measurability to functions that are not nec-
essarily nonnegative.
Definition 1.20.30.Let.X;S;/be a complete-finite measure space and letfW
X!Œ1;1.Wedefinethenonnegative partf
C
offby
f
C
.x/WD
´
f.x/iff.x/0;
0 iff.x/ <0:
Similarly, we define thenonpositive partf

offby
f

.x/WD
´
0 iff.x/0;
f.x/iff.x/ <0:

48. 28 Chapter 1 Preliminaries
Remark 1.20.31.The functionsf
C
andf

are nonnegative and
fDf
C
f

;jfjDf
C
Cf

:
Definition 1.20.32.Let.X;S;/be a complete-finite measure space and letfW
X!Œ1;1. We say thatfisS−measurable(or justmeasurable) if bothf
C
and
f

are measurable in the sense of Definition 1.20.28. We denote byM.X/the class
of all measurable functions onX.
Remark 1.20.33.Let.X;S;/be a complete-finite measure space. Then the char-
acteristic function
Eof a subsetEofXis measurable if and only if the setEis
measurable.
Theorem 1.20.34(Fatou lemma).Let.X;S;/be a complete−finite measure space
and let¹f
nº
1
nD1
be a sequence of nonnegative measurable functions onX.Then
Z
X
lim inf
n!1
fn.x/d.x/lim inf
n!1
Z
X
fn.x/d.x/:
Definition 1.20.35.Let.X;S;/be a complete-finite measure space and letfW
X!Œ1;1. We say thatfis−integrable(or justintegrable)ifboth
R
X
f
C
d
and
R
X
f

dare finite. We then define theintegraloffby
Z
X
f.x/d.x/WD
Z
X
f
C
.x/d.x/
Z
X
f

.x/d.x/: (1.20.2)
In such cases, we say that the integral offoverXconverges. We denote byL
1
.X;S;/
(or justL
1
.X/orL
1
./)thesetofall-integrable functions onX.
In the case when exactly one of the values
R
X
f
C
.x/d.x/,
R
X
f

.x/d.x/is
infinite, we again define the integral offby the formula (1.20.2), that is, as1or1,
and we say that the integral offoverXexists(although it does not converge). When
both values
R
X
f
C
.x/d.x/and
R
X
f

.x/d.x/are infinite, then we say that the
integral offoverXdoes not exist.
We shall now collect basic properties of integrable functions in the following propo-
sition.
Proposition 1.20.36.Let.X;S;/be a complete−finite measure space and let
f; gWX!Œ1;1anda; b2R.Then
(i)f2L
1
.X/holds if and only ifjfj2L
1
.X/and
ˇ
ˇ
ˇ
ˇ
Z
X
f.x/d.x/
ˇ
ˇ
ˇ
ˇ

Z
X
jf.x/jd.x/I
(ii)ifjf.x/jg.x/for a.e.x2Xandg2L
1
.X/,thenf2L
1
.X/;

49. Section 1.21 The Lebesgue measure and integral 29
(iii)iff.x/Dg.x/for a.e.x2Xandf2L
1
.X/,theng2L
1
.X/and
Z
X
f.x/d.x/D
Z
X
g.x/d.x/I
(iv)iff; g2L
1
.X/,then.fCg/2L
1
.X/and
Z
X
.af .x/Cbg.x//d.x/Da
Z
X
f.x/d.x/Cb
Z
X
g.x/d.x/I
(v)ifE2Sandf2L
1
.X/,thenf E2L
1
.X/and
Z
X
E.x/f .x/d.x/D
Z
E
f.x/d.x/:
Another important result concerning convergence is the following assertion.
Theorem 1.20.37(Lebesgue dominated convergence theorem).Let.X;S;/be a com−
plete−finite measure space and let¹f
nº
1
nD1
be a sequence of measurable functions
onXand letg2L
1
.X/be such that
jf
n.x/jg.x/
for alln2Nand almost allx2X. Assume that
lim
n!1
fn.x/Df.x/for a.e.x2X:
Then the following statements hold:
(i)f2L
1
.X/;
(ii)
R
X
f.s/d.x/Dlim n!1
R
X
fn.x/d.x/;
(iii) lim
n!1
R
X
jfn.x/f.x/jd.x/D0.
1.21 The Lebesgue measure and integral
In this section we apply the abstract general theory developed in Section 1.20 to the
particular case whenXD,whereis an open set inR
N
,N2N,Sis the-
algebra generated by allN-dimensional intervals andis the length function defined
by
..a
1;b1/.a N;bN//WD
N
Y
iD1
.biai/:
We denote the set of allN-dimensional intervals inR
N
byIN.
The restriction to an open setis temporary and will be abandoned in Chapter 6
and subsequent chapters, where more general measure spaces will be considered.

50. 30 Chapter 1 Preliminaries
Definition 1.21.1.TheLebesgue outer measure

is defined for everyA2exp./
by


.A/WDinf
´
1
X
nD1
.In\/II n2IN;Im\InD;form¤n; A
1
[
nD1
In
μ
:
The corresponding-algebra of

-measurable sets is called the-algebra ofLebesgue
measurable sets. The resulting measure is denoted byand is called theLebesgue
measure. The measure space.;B./; /is called theLebesgue measure space.
Proposition 1.21.2.The Lebesgue measure onR
N
,N2N, is a translation−invariant
(that is, ifA2Mandx2R
N
, then the setACxWD ¹y2R
N
;yx2Aºsatisfies
ACx2Mand.ACx/D.A/)−finite regular measure onB.R
N
/. In fact, it
is unique, up to multiplication by a positive constant.
Convention 1.21.3.Throughout the text, we shall often write
R

jf.x/jdxinstead
of
R

jf.x/jd.x/when no confusion can arise.
In the rest of this section we shall collect the most important assertions concern-
ing the Lebesgue integral. Some of them will be the “Lebesgue” versions of their
appropriate abstract counterparts from Section 1.20.
Theorem 1.21.4(Levi monotone convergence theorem).Let¹f
nº
1
nD1
be a sequence
of integrable functions on a measurable setR
N
such thatf n.x/f nC1.x/for
everyn2Nand almost allx2and that
Z

f1.x/dx>1:
Then, for almost allx2, the limit
f.x/Dlim
n!1
fn.x/
exists, the functionfis integrable and
Z

lim
n!1
fn.x/dxD
Z

f.x/dxDlim
n!1
Z

fn.x/dx:
Theorem 1.21.5(Lebesgue dominated convergence theorem).Let¹f
nº
1
nD1
be a se−
quence of measurable functions on a measurable setR
N
which converges for
almost allx2tof.x/. Suppose that there exists a functiongwith the finite
Lebesgue integral oversuch that
jf
n.x/jg.x/
for alln2Nand almost allx2.

51. Section 1.21 The Lebesgue measure and integral 31
Thenf n,n2N, andfhave finite integrals and
Z

f.x/dxDlim
n!1
Z

fn.x/dx:
Theorem 1.21.6(Fatou lemma).Let¹f
nº
1
nD1
be a sequence of measurable functions
which are nonnegative almost everywhere on.Then
lim inf
n!1
fn.x/
is integrable and
Z

lim inf
n!1
fn.x/dxlim inf
n!1
Z

fn.x/dx:
A generalization of Theorem 1.21.5 is given by the following result.
Theorem 1.21.7(Vitali).Let¹f
nº
1
nD1
be a sequence of functions with finite integrals
over a measurable setR
N
:Suppose that
lim
n!1
fn.x/Df.x/
for almost allx2and letfbe an almost everywhere finite function. Suppose that
the following condition is satisfied:
(P) for every">0there exists aı>0with the property: ifB; .B/ < ı;
then Z
B
jfn.x/jdx<"
for alln2N.
Then the functionfhas a finite integral overand
lim
n!1
Z

fn.x/dxD
Z

f.x/dx:
Very close to the preceding result is the following theorem.
Theorem 1.21.8(Vitali–Hahn–Saks).Let¹f
nº
1
nD1
be a sequence of functions with
finite integrals over a measurable setR
N
:Suppose that, for an arbitrary mea−
surable setE, the limit
lim
n!1
Z
E
fn.x/dx
exists and is finite.
Then the condition(P)inTheorem 1.21.7is satisfied.

52. 32 Chapter 1 Preliminaries
The following assertion is a direct consequence of Theorem 1.21.5.
Theorem 1.21.9(continuous dependence of the integral on a parameter).Let
R
N
be measurable and let.X; %/be a metric space. Letf.x;˛/be defined onX.
Suppose that
(i)for almost allx2; f.x;/is continuous onXI
(ii)for every˛2X; f .;˛/is measurable onI
(iii)there exists agwith a finite integral oversuch that
jf.x;˛/jg.x/
for all˛2Xand almost allx2:
Then the functionF, defined by
F.˛/WD
Z

f.x;˛/dx; ˛2X;
is continuous onX.
The following assertion is of great importance.
Theorem 1.21.10(derivative of the integral with respect to a parameter).Let
R
N
be measurable,1 a<b1and letf.x;˛/be defined on.a; b/.
Define the function
F.˛/WD
Z

f.x;˛/dx; ˛2.a; b/;
and suppose that
(i)F.˛/is finite for at least one˛2.a; b/I
(ii)for every˛2.a; b/,f.;˛/is measurable on;
(iii)the partial derivative
@f . x ; ˛ /
@˛
exists and is finite for every˛2.a; b/and almost everyx2;
(iv)there exists a functiongwith finite integral oversuch that
ˇ
ˇ
ˇ
ˇ
@f . x ; ˛ /
@˛
ˇ
ˇ
ˇ
ˇ
g.x/
for almost everyx2and all˛2.a; b/.

53. Section 1.21 The Lebesgue measure and integral 33
Then, for all˛2.a; b/, the integral
F.˛/D
Z

f.x; ˛/dx
is finite and
F
0
.˛/D
Z

@f . x ; ˛ /
@˛
dx:
To handle integrals over subsets ofR
N
we shall use the Fubini theorem (sometimes
in literature called the Fubini–Tonelli theorem).
Theorem 1.21.11(Fubini).Let
iR
Ni,iD1; 2, be measurable and setD

12:Letf.x;y/be integrable over:Then for almost allx2 1andy2 2
the integrals
Z
1
f.x;y/dxand
Z
2
f.x;y/dy
exist. Moreover,
Z

f.x;y/dxdyD
Z
1
Z
2
f.x;y/dy

dxD
Z
2
Z
1
f.x;y/dx

dy:
A characterization of measurable functions is given by the Luzin theorem.
Theorem 1.21.12(Luzin).LetR
N
be measurable; letfbe defined almost
everywhere on:Thenfis measurable onif and only if, for every">0,there
exists an open setM,.M/ < ";such that the restriction offtonMis
continuous onnM.
We shall now recall a usefulresult on the absolute continuous dependence of an
integral on the integration domain.
Theorem 1.21.13(absolute continuity of integral).Letfbe a function with a finite
Lebesgue integral overR
N
:Then, for every">0, there exists aıDı."/ > 0
such that for every measurable subsetEofwith.E/ < ?;we have
ˇ
ˇ
ˇ
ˇ
Z
E
f.x/dx
ˇ
ˇ
ˇ
ˇ
<":
Definition 1.21.14.Letbe a-additive set function defined on the family of all
Lebesgue measurable subsets of.Let.;/D0:We say thatisabsolutely contin−
uouswith respect to Lebesgue measureand write2AC ?,if
.E/D0implies.E/D0
for every-measurable setsE:

54. 34 Chapter 1 Preliminaries
Theorem 1.21.15(Radon–Nikodým).Let2AC ?be a finite set function. Then
there exists exactly one functionfwith a finite Lebesgue integral oversuch that
.E/D
Z
E
f.x/dx
for every Lebesgue measurable subsetE:
1.22 Modes of convergence
Throughout this section we shall assume that.X;S;/is a-finite complete mea-
sure space. As already noticed, the requirement of completeness is not too restrictive
because every measure space can be easily completed. The main reason why it is rea-
sonable to assume completeness is that in a complete space, the following implication
holds: iffandgare functions onX,fis-measurable and


.¹x2XIf.x/¤g.x/º/D0;
thengis also-measurable.
We shall study several types of convergence of a sequence of functions.
Definition 1.22.1.Letf
n,n2N,andfbe-measurable functions defined on a-
finite complete measure space.X;S;/.
(i) We say that the sequence¹f
nº
1
nD1
converges tofpointwiseonXif
lim
n!1
fn.x/Df.x/
for everyx2X. We writef
n!f.
(ii) We say that the sequence¹f
nº
1
nD1
converges tofuniformlyonXif for every
">0there exists ann
02Nsuch that for everynn 0and everyx2Xwe
have
jf
n.x/f.x/j<":
We writef
nf.
(iii) IfXis further endowed with a metric%, then we say that the sequence¹f
nº
1
nD1
converges toflocally uniformlyonXif¹f nº
1
nD1
converges tofuniformly on
each compact subsetKofX. We writef
n
locf.
(iv) We say that the sequence¹f
nº
1
nD1
converges tofuniformly up to small sets
onXif for every">0there exists anMX,.M/ < ", such that¹f
nº
1
nD1
converges uniformly tofonXnM.

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59. The Project Gutenberg eBook of The Alien

60. This ebook is for the use of anyone anywhere in the United States
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Title: The Alien
Author: Raymond F. Jones
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*** START OF THE PROJECT GUTENBERG EBOOK THE ALIEN ***

62. THE ALIEN
A Gripping Novel of Discovery and Conquest in Interstellar
Space
by Raymond F. Jones
A Complete ORIGINAL Book, UNABRIDGED
WORLD EDITIONS, Inc.
105 WEST 40th STREET
NEW YORK 18, NEW YORK
Copyright 1951
by
WORLD EDITIONS, Inc.
PRINTED IN THE U.S.A.
THE GUINN CO., Inc.
New York 14, N.Y.

63. [Transcriber's Note: Extensive research did not uncover any
evidence that the U.S. copyright on this publication was
renewed.]

64. Just speculate for a moment on the enormous challenge to
archeology when interplanetary flight is possible ... and relics are
found of a race extinct for half a million years! A race, incidentally,
that was scientifically so far in advance of ours that they held the
secret of the restoration of life!
One member of that race can be brought back after 500,000 years
of death....

65. That's the story told by this ORIGINAL book-length novel, which has
never before been published! You can expect a muscle-tightening,
sweat-producing, mind-prodding adventure in the future when you
read it!
Contents
CHAPTER ONE
CHAPTER TWO
CHAPTER THREE
CHAPTER FOUR
CHAPTER FIVE
CHAPTER SIX
CHAPTER SEVEN
CHAPTER EIGHT
CHAPTER NINE
CHAPTER TEN
CHAPTER ELEVEN
CHAPTER TWELVE
CHAPTER THIRTEEN
CHAPTER FOURTEEN
CHAPTER FIFTEEN
CHAPTER SIXTEEN
CHAPTER SEVENTEEN
CHAPTER EIGHTEEN

66. CHAPTER ONE
Out beyond the orbit of Mars the Lavoisier wallowed cautiously
through the asteroid fields. Aboard the laboratory ship few of the
members of the permanent Smithson Asteroidal Expedition were
aware that they were in motion. Living in the field one or two years
at a time, there was little that they were conscious of except the
half-million-year-old culture whose scattered fragments surrounded
them on every side.
The only contact with Earth at the moment was the radio link by
which Dr. Delmar Underwood was calling Dr. Illia Morov at Terrestrial
Medical Central.
Illia's blonde, precisely coiffured hair was only faintly golden against,
the stark white of her surgeons' gown, which she still wore when
she answered. Her eyes widened with an expression of pleasure as
her face came into focus on the screen and she recognized
Underwood.
"Del! I thought you'd gone to sleep with the mummies out there. It's
been over a month since you called. What's new?"
"Not much. Terry found some new evidence of Stroid III. Phyfe has
a new scrap of metal with inscriptions, and they've found something
that almost looks as if it might have been an electron tube five
hundred thousand years ago. I'm working on that. Otherwise all is
peaceful and it's wonderful!"
"Still the confirmed hermit?" Illia's eyes lost some of their banter, but
none of their tenderness.
"There's more peace and contentment out here than I'd ever
dreamed of finding. I want you to come out here, Illia. Come out for
a month. If you don't want to stay and marry me, then you can go
back and I won't say another word."

67. She shook her head in firm decision. "Earth needs its scientists
desperately. Too many have run away already. They say the
Venusian colonies are booming, but I told you a year ago that simply
running away wouldn't work. I thought by now you would have
found it out for yourself."
"And I told you a year ago," Underwood said flatly, "that the only
possible choice of a sane man is escape."
"You can't escape your own culture, Del. Why, the expedition that
provided the opportunity for you to become a hermit is dependent
on Earth. If Congress should cut the Institute's funds, you'd be
dropped right back where you were. You can't get away."
"There are always the Venusian colonies."
"You know it's impossible to exist there independent of Earth."
"I'm not talking about the science and technology. I'm talking about
the social disintegration. Certainly a scientist doesn't need to take
that with him when he's attempting to escape it."
"The culture is not to blame," said Illia earnestly, "and neither is
humanity. You don't ridicule a child for his clumsiness when he is
learning to walk."
"I hope the human race is past its childhood!"
"Relatively speaking, it isn't. Dreyer says we're only now emerging
from the cave man stage, and that could properly be called
mankind's infancy, I suppose. Dreyer calls it the 'head man' stage."
"I thought he was a semanticist."
"You'd know if you'd ever talked with him. He'll tear off every other
word you utter and throw it back at you. His 'head man' designation
is correct, all right. According to him, human beings in this stage
need some leader or 'head man' stronger than themselves for
guidance, assumption of responsibility, and blame, in case of failure

68. of the group. These functions have never in the past been developed
in the individual so that he could stand alone in control of his own
ego. But it's coming—that's the whole import of Dreyer's work."
"And all this confusion and instability are supposed to have
something to do with that?"
"It's been growing for decades. We've seen it reach a peak in our
own lifetimes. The old fetishes have failed, the head men have been
found to be hollow gods, and men's faith has turned to derision.
Presidents, dictators, governors, and priests—they've all fallen from
their high places and the masses of humanity will no longer believe
in any of them."
"And that is development of the race?"
"Yes, because out of it will come a people who have found in
themselves the strength they used to find in the 'head men.' There
will come a race in which the individual can accept the responsibility
which he has always passed on to the 'head man,' the 'head man' is
no longer necessary."
"And so—the ultimate anarchy."
"The 'head man' concept has, but first he has to find out that has
nothing to do with government. With human beings capable of
independent, constructive behavior, actual democracy will be
possible for the first time in the world's history."
"If all this is to come about anyway, according to Dreyer, why not try
to escape the insanity of the transition period?"
Illia Morov's eyes grew narrow in puzzlement as she looked at
Underwood with utter incomprehension. "Doesn't it matter at all that
the race is in one of the greatest crises of all history? Doesn't it
matter that you have a skill that is of immense value in these times?
It's peculiar that it is those of you in the physical sciences who are

69. fleeing in the greatest numbers. The Venusian colonies must have a
wonderful time with physicists trampling each other to get away
from it all—and Earth almost barren of them. Do the physical
sciences destroy every sense of social obligation?"
"You forget that I don't quite accept Dreyer's theories. To me this is
nothing but a rotting structure that is finally collapsing from its own
inner decay. I can't see anything positive evolving out of it."
"I suppose so. Well, it was nice of you to call, Del. I'm always glad to
hear you. Don't wait so long next time."
"Illia—"
But she had cut the connection and the screen slowly faded into
gray, leaving Underwood's argument unfinished. Irritably, he flipped
the switch to the public news channels.
Where was he wrong? The past year, since he had joined the
expedition as Chief Physicist, was like paradise compared with living
in the unstable, irresponsible society existing on Earth. He knew it
was a purely neurotic reaction, this desire to escape. But application
of that label solved nothing, explained nothing—and carried no
stigma. The neurotic reaction was the norm in a world so confused.
He turned as the news blared abruptly with its perpetual urgency
that made him wonder how the commentators endured the endless
flow of crises.
The President had been impeached again—the third one in six
months.
There were no candidates for his office.
A church had been burned by its congregation.
Two mayors had been assassinated within hours of each other.
It was the same news he had heard six months ago. It would be the
same again tomorrow and next month. The story of a planet
repudiating all leadership. A lawlessness that was worse than
anarchy, because there was still government—a government that

70. could be driven and whipped by the insecurities of the populace that
elected it.
Dreyer called it a futile search for a 'head man' by a people who
would no longer trust any of their own kind to be 'head man.' And
Underwood dared not trust that glib explanation.
Many others besides Underwood found they could no longer endure
the instability of their own culture. Among these were many of the
world's leading scientists. Most of them went to the jungle lands of
Venus. The scientific limitations of such a frontier existence had kept
Underwood from joining the Venusian colonies, but he'd been very
close to going just before he got the offer of Chief Physicist with the
Smithson Institute expedition in the asteroid fields. He wondered
now what he'd have done if the offer hadn't come.
The interphone annunciator buzzed. Underwood turned off the news
as the bored communications operator in the control room
announced, "Doc Underwood. Call for Doc Underwood."
Underwood cut in. "Speaking," he said irritably.
The voice of Terry Bernard burst into the room. "Hey, Del! Are you
going to get rid of that hangover and answer your phone or should
we embalm the remains and ship 'em back?"
"Terry! You fool, what do you want? Why didn't you say it was you? I
thought maybe it was that elephant-foot Maynes, with chunks of
mica that he thought were prayer sticks."
"The Stroids didn't use prayer sticks."
"All right, skip it. What's new?"
"Plenty. Can you come over for a while? I think we've really got
something here."

71. "It'd better be good. We're taking the ship to Phyfe. Where are
you?"
"Asteroid C-428. It's about 2,000 miles from you. And bring all the
hard-rock mining tools you've got. We can't get into this thing."
"Is that all you want? Use your double coated drills."
"We wore five of them out. No scratches on the thing, even."
"Well, use the Atom Stream, then. It probably won't hurt the
artifact."
"I'll say it won't. It won't even warm the thing up. Any other ideas?"
Underwood's mind, which had been half occupied with mulling over
his personal problems while he talked with Terry, swung startledly to
what the archeologist was saying. "You mean that you've found a
material the Atom Stream won't touch? That's impossible! The
equations of the Stream prove—"
"I know. Now will you come over?"
"Why didn't you say so in the first place? I'll bring the whole ship."
Underwood cut off and switched to the Captain's line. "Captain
Dawson? Underwood. Will you please take the ship to the vicinity of
Asteroid C-428 as quickly as possible?"
"I thought Doctor Phyfe—"
"I'll answer for it. Please move the vessel."
Captain Dawson acceded. His instructions were to place the ship at
Underwood's disposal.
Soundlessly and invisibly, the distortion fields leaped into space
about the massive laboratory ship and the Lavoisier moved
effortlessly through the void. Its perfect inertia controls left no
evidence of its motion apparent to the occupants with the exception

72. of the navigators and pilots. The hundreds of delicate pieces of
equipment in Underwood's laboratories remained as steadfast as if
anchored to tons of steel and concrete deep beneath the surface of
Earth.
Twenty minutes later they hove in sight of the small, black asteroid
that glistened in the faint light of the faraway Sun. The spacesuited
figures of Terry Bernard and his assistant, Batch Fagin, clung to the
surface, moving about like flies on a blackened, frozen apple.
Underwood was already in the scooter lock, astride the little
spacescooter which they used for transportation between ships of
the expedition and between asteroids.
The pilot jockeyed the Lavoisier as near as safely desirable, then
signaled Underwood. The physicist pressed the control that opened
the lock in the side of the vessel. The scooter shot out into space,
bearing him astride it.
"Ride 'em, cowboy!" Terry Bernard yelled into the intercom. He gave
a wild cowboy yell that pierced Underwood's ears. "Watch out that
thing doesn't turn turtle with you."
Underwood grinned to himself. He said, "Your attitude convinces me
of a long held theory that archeology is no science. Anyway, if your
story of a material impervious to the Atom Stream is wrong, you'd
better get a good alibi. Phyfe had some work he wanted to do
aboard today."
"Come and see for yourself. This is it."
As the scooter approached closer to the asteroid, Underwood could
glimpse the strangeness of the thing. It looked as if it had been
coated with the usual asteroid material of nickel iron debris, but
Terry had cleared this away from more than half the surface.

73. The exposed half was a shining thing of ebony, whose planes and
angles were machined with mathematical exactness. It looked as if
there were at least a thousand individual facets on the one
hemisphere alone.
At the sight of it, Underwood could almost understand the thrill of
discovery that impelled these archeologists to delve in the mysteries
of space for lost kingdoms and races. This object which Terry had
discovered was a magnificent artifact. He wondered how long it had
circled the Sun since the intelligence that formed it had died. He
wished now that Terry had not used the Atom Stream, for that had
probably destroyed the validity of the radium-lead relationship in the
coating of debris that might otherwise indicate something of the age
of the thing.
Terry sensed something of Underwood's awe in his silence as he
approached. "What do you think of it, Del?"
"It's—beautiful," said Underwood. "Have you any clue to what it is?"
"Not a thing. No marks of any kind on it."
The scooter slowed as Del Underwood guided it near the surface of
the asteroid. It touched gently and he unstrapped himself and
stepped off. "Phyfe will forgive all your sins for this," he said. "Before
you show me the Atom Stream is ineffective, let's break off a couple
of tons of the coating and put it in the ship. We may be able to date
the thing yet. Almost all these asteroids have a small amount of
radioactivity somewhere in them. We can chip some from the
opposite side where the Atom Stream would affect it least."
"Good idea," Terry agreed. "I should have thought of that, but when
I first found the single outcropping of machined metal, I figured it
was very small. After I found the Atom Stream wouldn't touch it, I
was overanxious to undercover it. I didn't realize I'd have to burn
away the whole surface of the asteroid."
"We may as well finish the job and get it completely uncovered. I'll
have some of my men from the ship come on over."

74. It took the better part of an hour to chip and drill away samples to
be used in a dating attempt. Then the intense fire of the Atom
Stream was turned upon the remainder of the asteroid to clear it.
"We'd better be on the lookout for a soft spot." Terry suggested. "It's
possible this thing isn't homogeneous, and Papa Phyfe would be very
mad if we burned it up after making such a find."
From behind his heavy shield which protected him from the stray
radiation formed by the Atom Stream, Delmar Underwood watched
the biting fire cut between the gemlike artifact and the metallic
alloys that coated it. The alloys cracked and fell away in large
chunks, propelled by the explosions of matter as the intense heat
vaporized the metal almost instantly.
The spell of the ancient and the unknown fell upon him and swept
him up in the old mysteries and the unknown tongues. Trained in the
precise methods of the physical sciences, he had long fought against
the fascination of the immense puzzles which the archeologists were
trying to solve, but no man could long escape. In the quiet, starlit
blackness there rang the ancient memories of a planet vibrant with
life, a planet of strange tongues and unknown songs—a planet that
had died so violently that space was yet strewn with its remains—so
violently that somewhere the echo of its death explosion must yet
ring in the far vaults of space.
Underwood had always thought of archeologists as befogged
antiquarians poking among ancient graves and rubbish heaps, but
now he knew them for what they were—poets in search of
mysteries. The Bible-quoting of Phyfe and the swearing of red-
headed Terry Bernard were merely thin disguises for their poetic
romanticism.
Underwood watched the white fire of the Atom Stream through the
lead glass of the eye-protecting lenses. "I talked to Illia today," he

75. said. "She says I've run away."
"Haven't you?" Terry asked.
"I wouldn't call it that."
"It doesn't make much difference what you call it. I once lived in an
apartment underneath a French horn player who practised eight
hours a day. I ran away. If the whole mess back on Earth is like a
bunch of horn blowers tootling above your apartment, I say move,
and why make any fuss about it? I'd probably join the boys on Venus
myself if my job didn't keep me out here. Of course it's different with
you. There's Illia to be convinced—along with your own conscience."
"She quotes Dreyer. He's one of your ideals, isn't he?"
"No better semanticist ever lived," Terry said flatly. "He takes the
long view, which is that everything will come out in the wash. I
agree with him, so why worry—knowing that the variants will iron
themselves out, and nothing I can possibly do will be noticed or
missed? Hence, I seldom worry about my obligations to mankind, as
long as I stay reasonably law-abiding. Do likewise, Brother Del, and
you'll live longer, or at least more happily."
Underwood grinned in the blinding glare of the Atom Stream. He
wished life were as simple as Terry would have him believe. Maybe it
would be, he thought—if it weren't for Illia.
As he moved his shield slowly forward behind the crumbling debris,
Underwood's mind returned to the question of who created the
structure beneath their feet, and to what alien purpose. Its black,
impenetrable surfaces spoke of excellent mechanical skill, and a high
science that could create a material refractory to the Atom Stream.
Who, a half million years ago, could have created it?
The ancient pseudo-scientific Bode's Law had indicated a missing
planet which could easily have fitted into the Solar System in the

76. vicinity of the asteroid belt. But Bode's Law had never been accepted
by astronomers—until interstellar archeology discovered the artifacts
of a civilization on many of the asteroids.
The monumental task of exploration had been undertaken more
than a generation ago by the Smithson Institute. Though always
handicapped by shortage of funds, they had managed to keep at
least one ship in the field as a permanent expedition.
Dr. Phyfe, leader of the present group, was probably the greatest
student of asteroidal archeology in the System. The younger
archeologists labeled him benevolently Papa Phyfe, in spite of the
irascible temper which came, perhaps, from constantly switching his
mind from half a million years ago to the present.
In their use of semantic correlations, Underwood was discovering,
the archeologists were far ahead of the physical scientists, for they
had an immensely greater task in deducing the mental concepts of
alien races from a few scraps of machinery and art.
Of all the archeologists he had met, Underwood had taken the
greatest liking to Terry Bernard. An extremely competent semanticist
and archeologist, Terry nevertheless did not take himself too
seriously. He did not even mind Underwood's constant assertion that
archeology was no science. He maintained that it was fun, and that
was all that was necessary.
At last, the two groups approached each other from opposite sides
of the asteroid and joined forces in shearing off the last of the
debris. As they shut off the fearful Atom Streams, the scientists
turned to look back at the thing they had cleared.
Terry said quietly, "See why I'm an archeologist?"
"I think I do—almost," Underwood answered.

77. The gemlike structure beneath their feet glistened like polished
ebony. It caught the distant stars in its thousand facets and cast
them until it gleamed as if with infinite lights of its own.
The workmen, too, were caught in its spell, for they stood silently
contemplating the mystery of a people who had created such
beauty.
The spell was broken at last by a movement across the heavens.
Underwood glanced up. "Papa Phyfe's coming on the warpath. I'll
bet he's ready to trim my ears for taking the lab ship without his
consent."
"You're boss of the lab ship, aren't you?" said Terry.
"It's a rather flexible arrangement—in Phyfe's mind, at least. I'm
boss until he decides he wants to do something."
The headquarters ship slowed to a halt and the lock opened,
emitting the fiery burst of a motor scooter which Doc Phyfe rode
with angry abandon.
"You, Underwood!" His voice came harshly through the phones. "I
demand an explanation of—"
That was as far as he got, for he glimpsed the thing upon which the
men were standing, and from his vantage point it looked all the
more like a black jewel in the sky. He became instantly once more
the eager archeologist instead of expedition administrator, a role he
filled with irritation.
"What have you got there?" he whispered.
Terry answered. "We don't know. I asked Dr. Underwood's assistance
in uncovering the artifact. If it caused you any difficulty, I'm sorry;
it's my fault."

78. "Pah!" said Phyfe. "A thing like this is of utmost importance. You
should have notified me immediately."
Terry and Underwood grinned at each other. Phyfe reprimanded
every archeologist on the expedition for not notifying him
immediately whenever anything from the smallest machined
fragment of metal to the greatest stone monuments were found. If
they had obeyed, he would have done nothing but travel from
asteroid to asteroid over hundreds of thousands of miles of space.
"You were busy with your own work," said Terry.
But Phyfe had landed, and as he dismounted from the scooter, he
stood in awe. Terry, standing close to him, thought he saw tears in
the old man's eyes through the helmet of the spaceship.
"It's beautiful!" murmured Phyfe in worshipping awe. "Wonderful.
The most magnificent find in a century of asteroidal archeology. We
must make arrangements for its transfer to Earth at once."
"If I may make a suggestion," said Terry, "you recall that some of
the artifacts have not survived so well. Decay in many instances has
set in—"
"Are you trying to tell me that this thing can decay?" Phyfe's little
gray Van Dyke trembled violently.
"I'm thinking of the thermal transfer. Doctor Underwood is better
able to discuss that, but I should think that a mass of this kind,
which is at absolute zero, might undergo unusual stresses in coming
to Earth normal temperatures. True, we used the Atom Stream on it,
but that heat did not penetrate enough to set up great internal
stresses."
Phyfe looked hesitant and turned to Underwood. "What is your
opinion?"
Underwood didn't get it until he caught Terry's wink behind Phyfe's
back. Once it left space and went into the museum laboratory, Terry
might never get to work on the thing again. That was the perpetual
gripe of the field men.

79. "I think Doctor Bernard has a good point," said Underwood. "I would
advise leaving the artifact here in space until a thorough
examination has been made. After all, we have every facility aboard
the Lavoisier that is available on Earth."
"Very well," said Phyfe. "You may proceed in charge of the physical
examination of the find, Doctor Underwood. You, Doctor Bernard,
will be in charge of proceedings from an archeological standpoint.
Will that be satisfactory to everyone concerned?"
It was far more than Terry had expected.
"I will be on constant call," said Phyfe. "Let me know immediately of
any developments." Then the uncertain mask of the executive fell
away from the face of the little old scientist and he regarded the find
with humility and awe. "It's beautiful," he murmured again,
"beautiful."

80. CHAPTER TWO
Phyfe remained near the site as Underwood and Terry set their crew
to the routine task of weighing, measuring, and photographing the
object, while Underwood considered what else to do.
"You know, this thing has got me stymied, Terry. Since it can't be
touched by an Atom Stream, that means there isn't a single
analytical procedure to which it will respond—that I know of,
anyway. Does your knowledge of the Stroids and their ways of doing
things suggest any identification of it?"
Terry shook his head as he stood by the port of the laboratory ship
watching the crews at work outside. "Not a thing, but that's no
criterion. We know so little about the Stroids that almost everything
we find has a function we never heard of before. And of course
we've found many objects with totally unknown functions. I've been
thinking—what if this should turn out to be merely a natural gem
from the interior of the planet, maybe formed at the time of its
destruction, but at least an entirely natural object rather than an
artifact?"
"It would be the largest crystal formation ever encountered, and the
most perfect. I'd say the chances of its natural formation are
negligible."
"But maybe this is the one in a hundred billion billion or whatever
number chance it may be."
"If so, its value ought to be enough to balance the Terrestrial
budget. I'm still convinced that it must be an artifact, though its
material and use are beyond me. We can start with a radiation
analysis. Perhaps it will respond in some way that will give us a
clue."

81. When the crew had finished the routine check, Underwood directed
his men to set up the various types of radiation equipment contained
within the ship. It was possible to generate radiation through almost
the complete spectrum from single cycle sound waves to hard
cosmic rays.
The work was arduous and detailed. Each radiator was slowly driven
through its range, then removed and higher frequency equipment
used. At each fraction of an octave, the object was carefully
photographed to record its response.
After watching the work for two days, Terry wearied of the
seemingly non-productive labor. "I suppose you know what you're
doing, Del," he said. "But is it getting you anywhere at all?"
Underwood shook his head. "Here's the batch of photographs. You'll
probably want them to illustrate your report. The surfaces of the
object are mathematically exact to a thousandth of a millimeter.
Believe me, that's some tolerance on an object of this size. The
surfaces are of number fifteen smoothness, which means they are
plane within a hundred-thousandth of a millimeter. The implications
are obvious. The builders who constructed that were mechanical
geniuses."
"Did you get any radioactive dating?"
"Rather doubtfully, but the indications are around half a million
years."
"That checks with what we know about the Stroids."
"It would appear that their culture is about on a par with our own."
"Personally, I think they were ahead of us," said Terry. "And do you
see what that means to us archeologists? It's the first time in the
history of the science that we've had to deal with the remains of a
civilization either equal or superior to our own. The problems are

82. multiplied a thousand times when you try to take a step up instead
of a step down."
"Any idea of what the Stroids looked like?"
"We haven't found any bodies, skeletons, or even pictures, but we
think they were at least roughly anthropomorphic. They were farther
from the Sun than we, but it was younger then and probably gave
them about the same amount of heat. Their planet was larger and
the Stroids appear to have been somewhat larger as individuals than
we, judging from the artifacts we've discovered. But they seem to
have had a suitable atmosphere of oxygen diluted with appropriate
inert gases."
They were interrupted by the sudden appearance of a laboratory
technician who brought in a dry photographic print still warm from
the developing box.
He laid it on the desk before Underwood. "I thought you might be
interested in this."
Underwood and Terry glanced at it. The picture was of the huge,
gemlike artifact, but a number of the facets seemed to be covered
with intricate markings of short, wavy lines.
Underwood stared closer at the thing. "What the devil are those? We
took pictures of every facet previously and there was nothing like
this. Get me an enlargement of these."
"I already have." The assistant laid another photo on the desk,
showing the pattern of markings as if at close range. They were
clearly discernible now.
"What do you make of it?" asked Underwood.
"I'd say it looked like writing," Terry said. "But it's not like any of the
other Stroid characters I've seen—which doesn't mean much, of
course, because there could be thousands that I've never seen. Only

83. how come these characters are there now, and we never noticed
them before?"
"Let's go out and have a look," said Underwood. He grasped the
photograph and noted the numbers of the facets on which the
characters appeared.
In a few moments the two men were speeding toward the surface of
their discovery astride scooters. They jockeyed above the facets
shown on the photographs, and stared in vain.
"Something's the matter," said Terry. "I don't see anything here."
"Let's go all the way around on the scooters. Those guys may have
bungled the job of numbering the photos."
They began a slow circuit, making certain they glimpsed all the
facets from a height of only ten feet.
"It's not here," Underwood agreed at last. "Let's talk to the crew that
took the shots."
They headed towards the equipment platform, floating in free space,
from which Mason, one of the Senior Physicists, was directing
operations. Mason signaled for the radiations to be cut off as the
men approached.
"Find any clues, Chief?" he asked Underwood. "We've done our best
to fry this apple, but nothing happens."
"Something did happen. Did you see it?" Underwood extended the
photograph with the mechanical fingers of the spacesuit. Mason held
it in a light and stared at it. "We didn't see a thing like that. And we
couldn't have missed it." He turned to the members of the crew.
"Anyone see this writing on the thing?"
They looked at the picture and shook their heads.
"What were you shooting on it at the time?"
Mason glanced at his records. "About a hundred and fifty
angstroms."

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