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 (r, x) = 0 implies r = 0; (r, x) 20 for all x e H; (1, y) = (y, x); for any a € C, (ax + y, w) = a (r, 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"-,, is a Hilbert space. %D (c) Show that C" with (r, y) = E=1 kak is a Hilbert space. %3D (d) Show that H {S € C[r] : deg s< 5} with (f, g) = S(t) g(t) dt is a Hilbert space. (e) Show that H = {ƒ € C[r]: deg f < 5} with (S, 9) E-o S(k)g(k) is a Hilbert Lk=0 space.
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 (r, x) = 0 implies r = 0; (r, x) 20 for all x e H; (1, y) = (y, x); for any a € C, (ax + y, w) = a (r, 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"-,, is a Hilbert space. %D (c) Show that C" with (r, y) = E=1 kak is a Hilbert space. %3D (d) Show that H {S € C[r] : deg s< 5} with (f, g) = S(t) g(t) dt is a Hilbert space. (e) Show that H = {ƒ € C[r]: deg f < 5} with (S, 9) E-o S(k)g(k) is a Hilbert Lk=0 space.
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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3A please
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d040c73-5bcf-4909-aa2b-abaad050d675%2F9b36d7c9-5dcb-4e04-891a-557c7497b380%2F1o8bsej_processed.jpeg&w=3840&q=75)
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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