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.