Math 241A Fall 2013 HOME WORK
A paper for reference. Examples will be drawn from it.
Chapter 3 Banach Spaces
Optional in 2013
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 ??
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 ??
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.