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 the
sequence if necessary, we may assume that An < 1 f
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 exists a linear operator B, so that B g , and g( ) = B,
f
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
function of (, 1). Clearly, the sequences (Tn f )nN and (Tn
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| = sup 1 |A|, .
(b) It remains to show that S1 is complete
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(0, c). Note rst that, for A E , the following are
equiv
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 C0 (R) and f Cb (R), let (f ) = f (here,
g = suptR |g(
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 nite subcover, consisting of the sets xi + V (1 i n).
A f
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 subspace of Cb (R), consisting of the functions conver
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 , . . . are self-adjoint operatorson a Hilbert space H, and
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 operator. Prove that there exists n N
so that ker(I T )m = ker
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 , A2 , . . . L(H) (H is a Hilbert space) are such that A
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, and T, T1 , T2 . . . L(X) satisfy limn T
Tn = 0. Suppo
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), 22.
A. (a) Prove that, if T 0, and k N, then T k 0.
(
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 functions on [0, 1], which have
continuous derivatives of all
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 measure on cfw_0, 1 by setting (0) = (1) = 1/2, and let
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, talk to me rst.
VI.16.
VII.31.
A. Suppose X is a vect
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 with
the topology (X , X ); can consider the real case