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A finite-dimensional Hilbert space is a finite-dimen with an inner product (-, ·). This means that (x,! O implie

A finite-dimensional Hilbert space is a finite-dimen with an inner product (-, ·). This means that (x,! O implie

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 24EQ
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plz solve the question (a) with explanation i will give you upvote
3. A finite-dimensional Hilbert space is a finite-dimensional complex vector space H together with an inner product (, -). This means that (r, y) €C for all r, y E H and (x, x) = 0 implies r = 0; (x, r) 2 0 for all r € H; (x, y) = (y, a); • for any a e C, (ax + y, w) = a (x, w) + (y, w). (a) Show that (r, ay + w) = a(r, y) + (r, w) for all r, y, w E H, a € C. (b) Show that C" with (r, y) =E" ,, is a Hilbert space. %3D %3D (c) Show that C" with (r, y) = E- kark is a Hilbert space. %3D (d) Show that H = {f € C[x] : degƒ < 5} with (f, 9) = So s(1) g(t) dt is a Hilbert space. (e) Show that H = {ƒ € C[r] : deg f < 5} with (f, g) = E-o S(k)g(k) is a Hilbert %3D space.
Transcribed Image Text:3. A finite-dimensional Hilbert space is a finite-dimensional complex vector space H together with an inner product (, -). This means that (r, y) €C for all r, y E H and (x, x) = 0 implies r = 0; (x, r) 2 0 for all r € H; (x, y) = (y, a); • for any a e C, (ax + y, w) = a (x, w) + (y, w). (a) Show that (r, ay + w) = a(r, y) + (r, w) for all r, y, w E H, a € C. (b) Show that C" with (r, y) =E" ,, is a Hilbert space. %3D %3D (c) Show that C" with (r, y) = E- kark is a Hilbert space. %3D (d) Show that H = {f € C[x] : degƒ < 5} with (f, 9) = So s(1) g(t) dt is a Hilbert space. (e) Show that H = {ƒ € C[r] : deg f < 5} with (f, g) = E-o S(k)g(k) is a Hilbert %3D space.
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