Inner product space

What is Inner Product Space?
Properties of Inner product space

1. GROUP MEMBERS: Iqra Bibi 59 Shehar Bano Annum Waheed Maryam Abrar Abbasi Sana Farooq BS( Hons ) Mathematics 3rd

2. Inner Product Space

3. Inner Product An Inner Product on a real space V is a function that associates a number, denoted< u,v > , with each pair of vectors u and v of V. This function has to satisfy the following condition s for u, v, w and scalar c.

4. PROPERTIES <u, v>=<v, u> ( symmetry axioms) <u+v, w>=<u,w> + <v,w>( additive axiom) < cu,v >=c < u,v > ( homogeneity axiom) < v,v > ≥ 0 and < v,v >=0 iff v=0 ( positivity axiom)

5. Inner Product Space Definition: An inner product space is a vector space with an additional structure called an Inner Product. This additional structure associate each pair of vectors in the space with a scalar quantity known as the inner product of the vectors.

6. Properties of Inner Product Space If u and v are vectors in a real inner product space V, and if k is a scalar, then: ( a )||v||≥0 with equality if and only if v=0. ( b ) || kv ||=|k| ||v||. ( c ) d( u,v )=d( v,u ) ( d ) d( u,v ) ≥ 0with equality if and only if u=v .

7. Weighted Euclidean Inner Products Let and be vectors in . Verify that the weighted Euclidean inner product Satisfy the four inner product axioms.

8. Example Let u=( ) and v=( ) be vectors in Axiom 1 : Interchanging u and v in formula does not change the sum on the right side, so Axiom 2: If ,then

9. Examples: Axiom 3: Axiom 4: with equality iff that is,if and only if v=0

10. Unit circle and spheres in inner product spaces Unit Sphere: If V is an inner product space, then the set of points in V that satisfy is called the unit sphere or sometimes the unit circle in V.

11. NORMS OF A VECTOR IN C [a, b] If C [a , b] has the inner product was defined in Example 10,then the norms of a function= f(x) relative to inner product is and the unit sphere in the space consist of all functions f in C [a ,b]that satisfy the equation ​ The arc length of a curve y=f(x) over an interval [a, b] is given by the formula

12. Theorem: If u,v and w are vectors in a real inner product spaces V and if k is a scalar, then : (0,v)=(v,0)=0 ( u,v+w )=( u,v )+(u,w) ( u,v -w) =( u,v )-(u,w) (u-v ,w) =(u,w)_(v,w) k(u ,v)=(u , kv )   The following example illustrate show theorem 6.1.2 and the defining properties of inner products can be used to perform algebraic computations with innexample , you will find it instructive to justify the steps. througher products as you read

13. Example: Calculating with inner products (u-2v ,3u+4v) =(u,3u+4v)-(2v,3u+4v) = (u,3u)+(u,4v)-(2v,3u)-(2v,4v) =3(u, u)+4(u, v)-6(v, u)-8(v, v) =||3||^2 +4( u,v )-6( u,v )-8||v||^2 =||3||^2 -2( u,v )- 8||v||^2 ◄