Let X be an inner product space with the inner product given by <, > . For x € X,define the function ||-· ||: X → K given by ||x|| = < x, − x >1/2, the non negativesquare root of < x, x>. Show that ||· || : X → K defines a norm on X and|< (x, y) >| ≤ ||x|| ||y|| for all x, y € X. Also show that for all x, yɛ X,|x + y² + ||x - y² = 2|x||² +|y||²).

Given : x=x , x12 To Prove : (a) ·:X→K is Norm : (b) x+y2+x-y2=2 x2+y2

Answered: Let X be an inner product space with…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let V be the vector space of polynomials in & over R of degree <3 and define an inner product f. g = f₁¹ f(x)g(x)dxæ for all f, g € V. You may assume that this makes V into an inner product space. Define the linear map a: V→ V by a(f) (x) = f(x)+xƒ'(x) where the f' denotes differentiation. Which of the following holds with respect to this inner product? Select one: O a is self-adjoint O None of the others apply Ⓒa* = I-a, where I denotes the identity map O a is `anti-self adjoint' meaning a* = -a a is orthogonal
2. Let [0, 1] := {ƒ : [0,1] → R : f is a bounded function on [0, 1]}. Let ||f|| sup f(x)| x= [0,1] for fel [0, 1]. Show that ( [0, 1], || - ||) is a Banach space.
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(b) space and X+{ 0 }. Show that B< (X, Y) is a Banach space. Let X be a normed space and Y be a Banach
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Verify that the set C[a, b] of all continuous real-valued functions defined on the interval a ≤ x ≤ b is a vector space, with addition and numerical multiplication defined by (f+g)(x) = f(x) + g(x) and (tf)(x) = tf(x).
Consider the inner product of two functions defined by {ƒ, g) = f w(x) f(x)g(x) dx a where w(x) is a weight function. (a) Show the set of the functions {1, x, 5x² - 1} is orthogonal with re- - spect to the weight function w(x) = 1 − x² on the interval [−1, 1]. (b) Make the set an orthonormal set of functions.
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1. Let f : R" → R" be a function with the property that for any v, w E R" it holds that f(v) · f(w) = V· W. I.e. f preserves the inner product on the vector space R". Show that f is a linear transfor- mation.
Let V be the vector space of polynomials in over R of degree ≤ 3 and define an inner product f . g = f¹ f(x)g(x)dx for all f, g € V. You may assume that this makes V into an inner product space. Define the linear mapa: V → V by a(ƒ)(x) = f(x) + xƒ'(x) where the f' denotes differentiation. Which of the following holds with respect to this inner product? Select one: O O O a is self-adjoint a is orthogonal a is 'anti-self adjoint' meaning a* = -α a = I-a, where I denotes the identity map None of the others apply
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