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Mathematics 418b/507b
Winter 2005
Homework 6.
Due Monday March 14
1. Textbook, # 7.21 (p. 85)
2. Textbook, # 7.38 (p. 87)
3. Textbook # 8.5 (p.103)
4. Prove that the closed unit ball B = cfw_|x| 1 in a normed space X is compact if and only if X
is nite-di
Mathematics 418b/507b
Winter 2005
Homework 7.
Due Monday March 21.
1. Textbook, # 8.12 (p. 103)
2. Textbook, # 8.13 (p. 103)
3. Find the spectrum of the Volterra operator V : L2 (0, 1) L2 (0, 1) given by V (x(t) =
t
0 x(s)ds.
4. (* for 507b only) Let K :
Mathematics 418b/507b
Winter 2005
Homework 8.
Due Wednesday, March 30.
1. Let An , A be bounded linear operators on a Hilbert space H. Prove
(i) if (An f, g) (Af, g) uniformly for |g| = 1, then |An f Af | 0, as n ;
(ii) if |An f Af | 0 uniformly for |f |
Mathematics 418b/507b
Winter 2005
Homework 5.
Due Wed, March 2.
1. Textbook, # 7.7 (p. 84);
2. Textbook, # 7.16 (p. 85);
3. Textbook, # 7.29 (p. 86)
4. Prove that if A : E E is a bounded linear operator of a normed space E, then the transpose
linear opera
Mathematics 418b/507b
Winter 2005
Homework 4.
Due Wed, February 16.
1. Textbook, # 5.1(b) (p. 55);
2. Textbook, # 6.3 (p. 64);
3. Textbook, # 6.7 (p. 65)
4. Let H be a Hilbert space, let W be a subspace, and let L : W C be a bounded linear functional.
Sho
Mathematics 418b/507b
Winter 2005
Homework 1.
Due Wednesday, January 19.
1. Prove that two metrics d and on a set X are equivalent if and only if given x X and
there exists a > 0 such that for any y X
>0
d(x, y) < = (x, y) <
and
(x, y) < = d(x, y) <
2. Pr
Mathematics 418b/507b
Winter 2005
Homework 2.
Due Wednesday, January 26.
1. Textbook, # 2.3 (p. 19):
2. Textbook, # 2.6 (p. 19):
3. Textbook, # 2.13 (p. 20):
4. Textbook, # 3.7 (p. 28);
5. (a) Let V be a vector space and W its linear subspace. Dene x y if
Mathematics 418b/507b
Winter 2005
Homework 3.
Due Monday, February 7.
1. Let H and K be nite dimensional linear spaces over C (of the same dimension). Prove that
t
f : H K is unitary if and only if f can be represented by a unitary matrix A, i.e. A = A1 ,
Math 418b/507b, Winter 2005
Introduction to Hilbert Spaces
Syllabus
Instructor: Rasul Shakov, MC 112. E-mail: [email protected] (emails will be answered within
48 hours), Oce Hours: Thursday 10 AM - 12 PM.
Textbook: An Introduction to Hilbert Space by N. Yo