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(r, x) = 0 implies z = 0; • (r, x) 20 for all z e H; • (r, y) = (y, x); • for any a € C, %3D (ar + y, w) = a (a a) Show that (r, ay+ w) = a(x, y) + (r, w) for p) Show that C" with (r, y) = E-1",, is a F =) Show that C" with (r, y) =E-, kaLTk is a %3D %3D =D1 %3D %=D1 1) Show that H = {S € C[r]: deg f < 5} w %3D

(r, x) = 0 implies z = 0; • (r, x) 20 for all z e H; • (r, y) = (y, x); • for any a € C, %3D (ar + y, w) = a (a a) Show that (r, ay+ w) = a(x, y) + (r, w) for p) Show that C" with (r, y) = E-1",, is a F =) Show that C" with (r, y) =E-, kaLTk is a %3D %3D =D1 %3D %=D1 1) Show that H = {S € C[r]: deg f < 5} w %3D

Advanced Engineering Mathematics
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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3. A finite-dimensional Hilbert space is a finite-dimensional complex vector space H together with an inner product (-, -). This means that (r, y) E C for all r, y E H and (r, x) = 0 impliesr = 0; (z, x) 20 for all r € H; (1, y) = (y, a); • for any a e C, (ar + y, w) = a (x, w) + (y, w). (a) Show that (r, ay + w) = a{x, y) + (r, w) for all r, y, w e H, a E C. (b) Show that C" with (r, y) = E" ,T, is a Hilbert space. (c) Show that C" with (r, y) =E kr is a Hilbert space. Lk=1 (d) Show that H = {f € C[x] : deg f < 5} with (f,g) = L S(t) g(t) dt is a Hilbert %3D space. (e) Show that H = {ƒ € C[r] : deg f < 5} with (f, g) = E-o S(k)g(k) is a Hilbert 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) E C for all r, y E H and (r, x) = 0 impliesr = 0; (z, x) 20 for all r € H; (1, y) = (y, a); • for any a e C, (ar + y, w) = a (x, w) + (y, w). (a) Show that (r, ay + w) = a{x, y) + (r, w) for all r, y, w e H, a E C. (b) Show that C" with (r, y) = E" ,T, is a Hilbert space. (c) Show that C" with (r, y) =E kr is a Hilbert space. Lk=1 (d) Show that H = {f € C[x] : deg f < 5} with (f,g) = L S(t) g(t) dt is a Hilbert %3D space. (e) Show that H = {ƒ € C[r] : deg f < 5} with (f, g) = E-o S(k)g(k) is a Hilbert space.
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