Math 241A Fall 2013 HOME WORK

TEXT: Conway "Functional Analysis"     Edition 2

Please turn in ths HW the beginning of section ??
Please tell me what you think of the Homework.         I NEED FEEDBACK!!

Lecture 1. Defs of Banach space and Banach Algebra.
Examples: C[0,1] and L^p
Ch 1 Hilbert Space.
p6 sec 1   ex 3 Show that H is a preHilbert space. You do not need to prove it is complete.
Chapter 1         p6 sec 1   ex 6         .
  p11 sec 2   ex 1,2, 3
  p13 sec 3   ex 5
  p18 sec 4   ex 13, 19
  p23 sec 5   ex 2, 3

Chapter 2
Operators on Hilbert space
sec 1 Great Examples of Operators   ex 2
Section 2, Adjoints ex. 6, 11, 16

HW 1 Turn in the above on ??

Chapter 2
Section 1, ex. 1, 3, 5, 8
Section 3 Projections , ex. 6, 11
Section 4 Compact Operators, ex. 1, 2, 4, 5
Section 5, Diagonalizing Compact SelfAdj Operators, ex. 1

HW 2 Turn the above in on ??

A paper for reference. Examples will be drawn from it.

Chapter 3 Banach Spaces

Bill already lectured on sec 1, 2 in the early part of the course while discussing Hilbert Space.
Section 1, ex. 3, 13
Section 2, ex. 5, 6
Section 3, ex. 1
Section 4, Quotient Spaces ex. 1, 6
Section 5, Linear Functionals , ex. 2

HW 3 Turn the above in on ??

Optional in 2013

Section 6, HB Theorem , READ IT
Section 7, Banach Limits, p. 83 SKIPPED
Section 9, Ordered Vector Spaces, p.88, ex. 4, 6, 7, 8, 9
Order1.     As defined in the Banach Limit section 3.7 take
          V to be l^\infty   and   S = sequences which posess a limit C= all nonnegative sequences in V.
What are (all of) the order units in S?

Ch 4 Section 3 Separating Hyperplanes   Ex 2, 7   and   Ex 10 with TVS replaced by Banach Space.
End Optional

Spectral Theory
If you have not done so yet do this:
S1.   Suppose $M$ is multiplication by x on L^2[0,1, u] where u is a probability measure supported on [0,1]. For which such measures u is M a compact operator?

S2. Suppose A1 A2 A3 are commuting self adjoint operators on a seperable Hilbert space H.
Suppose the spectrum of Aj is contained in [-2,2] for each j.
Let R:= [-2,2]^3
The following is true and you may assume it:
If p(x) is positive on R, then p(A) is PsD. Here x =(x1,x2,x3).

PROBLEMs on cyclic vectors:
C0. Say exactly which symmetric matrices have a cyclic vector
C1. What is a natural notion of cyclic vector for A1 A2 A3
C2. Assuming that they have a cyclic vector, what is a natural spectral representation for A1 , A2, A3?
C3. Prove it

HW 4   Discuss the spectral theory problems above with the Prof in Week 10 of the quarter.