SOLUTIONS FOR HOMEWORK 4
Chapter VI
14. Recall that, for 0 T I, we have T = I + ck (I T )k , with
k=1
ck < 0 for each k, and (ck ) 1.
k=1
(a) By scaling, we can assume that A < 1. Deleting the head of
SOLUTIONS FOR HOMEWORK 6
Chapter VI
30. (e) Then map H H C : (, ) g( ) is a bounded sesquilinear
form. Indeed, ( ) ( ) = 2 , hence | | = , which
leads to 1 = . Thus, |g( )| g . Consequently,
there exi
SOLUTIONS FOR HOMEWORK 3
Chapter VI
5. (a) Clearly, Tt is an isometry. limt Tt does not exist in the norm topology,
or in the strong operator topology. Indeed, take f = (0,1) (the characteristic
funct
SOLUTIONS FOR HOMEWORK 5
Chapter VI
23. (a) We know that A = |A| . Suppose is a norm 1 vector. Pick an
orthonormal basis (i ), containing . Then A 1 = i |A|i , i |A|, .
Due to self-adjointness, |A| =
SOLUTIONS FOR HOMEWORK 2
Chapter V
56. (a) Suppose E is a Frechet space, whose topology is determined by a
translation invariant metric d. For e E and c 0, dene B(e, c) = cfw_x
E : d(e, x) c = e + B(
SOLUTIONS FOR MIDTERM
A. In this problem, we denote by Cb (R) the space of bounded continuous functions on R. Let C0 (R) be the subspace of Cb (R), consisting of the functions converging to 0 at . For
SOLUTIONS FOR HOMEWORK 1
Chapter V
4 (a) Suppose U is a compact neighborhood of 0. By denition, there exists
an open set V , so that 0 V U. The sets x + V /2 (x U) form an open
cover of U. It has a ni
MATH 260B MIDTERM
The solutions are due on Friday, February 10, at the beginning of the class.
A. In this problem, we denote by Cb (R) the space of bounded continuous functions on R. Let C0 (R) be the
HOMEWORK 7
Problems assigned up to and including Friday, 03/09. The solutions are due
on Friday, 03/16.
Assigned Friday, 03/09:
Chapter VII: 7, 20 (f (x) = n for x 1/n), 24.
A. Suppose A, A1 , A2 , .
HOMEWORK 5
Problems assigned up to and including Friday, 02/17. The solutions are due
on Friday, 03/02.
Assigned Monday, 02/13:
Chapter VI: 23, 25, 26(b), 28(c).
A. Suppose T B(X) is a compact operato
HOMEWORK 6
Problems assigned up to and including Friday, 03/02. The solutions are due
on Friday, 03/09.
Assigned Wednesday, 02/29:
Chapter VI: 30(e,f), 46(b).
Chapter VII: 8(c), 12.
A. Suppose A, A1 ,
HOMEWORK 3
Problems assigned up to and including Friday, 01/27. The solutions are due
on Friday, 02/03.
Assigned Wednesday, 01/25:
Chapter VI: 5(a), 6(c,e), 8, 10.
A. (a) Suppose X is a Banach space,
HOMEWORK 4
Problems assigned up to and including Friday, 02/03. The solutions are due
on Friday, 02/17.
Assigned Friday, 02/03:
Chapter VI: 14, 15, 18(a) (the projections are assumed to be orthogonal)
HOMEWORK 2
Problems assigned up to and including Friday, 01/20. The solutions are due
on Friday, 01/27.
Assigned Wednesday, 01/18:
Chapter V: 56(a).
A. Consider the space C [0, 1] of continuous functi
SOLUTIONS FOR HOMEWORK 7
Chapter VII
7. Let C = cfw_0, 1N be the Cantor set, which we identify with the ternary
i
subset of [0, 1], via the isomorphism : (si )
i=1
i (2/3) si . Dene the
probability m
MATH 260B FINAL
The solutions are due on Friday, March 23, 1 pm. Place the hard copies of your
solutions into my mailbox. If you plan to be out of town, and want to submit the
solutions electronically
HOMEWORK 1
Problems assigned up to and including Friday, 01/13. The solutions are due
on Friday, 01/20.
Assigned Wednesday, 01/11:
Chapter V: 4, 5 (apply Theorem V.4(c) to some subsets of X , equipped