Metric-space-1-Calculus with Analytic Geometry

Math related

1. Metric space

2. Metric space A set whose elements we shall call points , is said to be a metric space if any two points p and q of there is associated a real number p to q, such that if ; for any  

3. REMARKS A metric space is a pair (X, d), where X is a set and d is a metric defined on X. The metric is often regarded as a distance function. The usual metric on R is the one given by d(x, y) = |x − y|. A metric can be used to define limits and continuity of functions. In fact, the ε-δ definition for functions on R can be easily adjusted so that it applies to functions on an arbitrary metric space

4. Discrete metric space Let be a non empty set. A map is defined as if is called discrete metric on  

5. Examples of metric in   The usual metric in is the Euclidean metric defined by The metric is defined by the formula  

6. The discrete metric on a nonempty set is defined by letting  

7. Open ball Suppose (X, d) is a metric space and let x ∈ X be an arbitrary point. The open ball with centre x and radius r > 0 is defined as The open balls in R are the open intervals B(x, r) = (x − r, x + r). The open interval (a, b) has center (b + a)/2 and radius (b − a)/2. If the metric on X is discrete, then B(x, 1) = {x} for all x ∈ X. The open ball B(0, 1) in X = [0, 2] is given by B(0, 1) = [0, 1).  

8. Examples Find and draw the following open ball .  

9. Answers  

11. Bounded set Let (X, d) be a metric space and A ⊂ X. We say that A is bounded, if there exist a point x ∈ X and some r > 0 such that A ⊂ B(x, r).

12. Example Let be a metric space Show that is bounded Is bounded?  

13. Solution:   Given Solution: Draw a circle with center at the open interval (3,6) with Let  

14. Open set Given a metric space (X, d), we say that U ⊂ X is open in X if, for each point x ∈ U there exists ε > 0 such that B(x, ε) ⊂ U. In other words, each x ∈ U is the center of an open ball that lies in U.

15. Example Let Show that is open Is open? Which of the following sets is open?  

16. Given We show that U is open by showing that There is a ball B(x, ε) ⊂ U. There are balls with such that B(x, ε) ⊂ U.  

17. Open set Let S be a subset of , S is said to be an open set if each point of S is an interior point of S Example Let No point of S is an interior of S, hence S is not an open set  

18. In a metric space, an open set is a set that contains all points that are sufficiently near to a given point In terms of distance, a set is open in if whenever it contains a number , it also contains all numbers sufficiently close to   OPEN SET Let be a metric space and . G is said to be open if for every such that  

19. example Consider a metric space where d is the usual distance Let Let be any arbitrary point Let   Now we prove Let By definition & Therefore Hence, is an open set  

20. Convergent sequence Let (X, d) be a metric space. We say that a sequence of points of X converges to the point x ∈ X if, given any ε > 0 there exists an integer N such that < ε for all n ≥ N   Introduction A sequence in is said to be convergent and converges to if for given > 0  

21. When a sequence converges to a point x, we say that x is the limit of the sequence and we write as n → ∞ or simply A sequence of points in converges if and only if each of the components converges in R  

22. The sequence { } n∈N converges to v if the sequence of numbers {d(v, )} n∈N converges to zero. The sequence { } n∈N converges to v if for every ε > 0 the open ball Bv,ε contains all but a finite number of terms of the sequence.  

23. remarks The limit of a sequence in a metric space is unique. In other words, no sequence may converge to two different limits.

24. Closed set Suppose (X, d) is a metric space and let A ⊂ X. We say that A is closed in X, if its complement X − A is open in X.

25. Example Given Determine whether or not is closed Solution: Show that is open Since is a subset of and is open Then is closed  

26. Main facts about closed set If a subset A ⊂ X is closed in X, then every sequence of points of A that converges must converge to a point of A. Both ∅ and X are closed in X. Finite unions of closed sets are closed. Arbitrary intersections of closed sets are closed.

27. Continuity in metric spaces

28. Theorems involving continuity Composition of continuous function Suppose f : X → Y and g : Y → Z are continuous functions between metric spaces. Then the composition g ◦ f : X → Z is continuous Continuity and sequences Suppose f : X → Y is a continuous function between metric spaces and let be a sequence of points of X which converges to x ∈ X. Then the sequence must converge to f(x).   CONTINUITY AND OPEN SETS A function f : X → Y between metric spaces is continuous if and only if f −1 (U) is open in X for each set U which is open in Y .

29. Convergence of a sequence A sequence converges to if for all there exists some N such that for all Let . The - neighborhood of a point is the interval Example.  

30. examples For example {1/n : n ∈ N} converges in R 1 and diverges in (0, ∞). consider the following sequence of complex number (i.e. ) If then ; the range is infinite, and the sequence is bounded. If then the sequence is divergent; the range is infinite, and the sequence is unbounded. If then the sequence converges to 1, is bounded, and has infinite range. If the sequence is divergent, is bounded and has finite range. If (n = 1, 2, 3, . . .) then converges to 1is bound  

31. Cauchy sequence Let (X, d) be a metric space. A sequence of points of X is called Cauchy if, given any ε > 0 there exists an integer N such that for all m, n ≥ N. Note: In a metric space every convergent sequence is Cauchy and every Cauchy sequence is bounded  

32. Complete Metric Space A metric space (X, d) is called complete if every Cauchy sequence of points of X actually converges to a point of X. Cauchy sequence with convergent subsequence Suppose (X, d) is a metric space and let be a Cauchy sequence in X that has a convergent subsequence. Then converges itself Completeness of Every sequence in R which is monotonic and bounded converges. Bolzano- Weierstrass theorem: Every bounded sequence in R has a convergent subsequence. The set R of all real numbers is a complete metric space.