Math 255A Functional Analysis
Dimitri Shlyakhtenko MS7901
Textbook: Conway, "A course in
Functional Analysis" (Springer Graduate Texts in Mathematics, 2nd
Edition)
Grading policy: There will be
a weekly homework (due Fridays; Wednesdays if Friday falls on a
Holiday).
Material: The aim of the course
is to go over the basics of single operator theory. We'll start
by talking about Hilbert spaces and Banach spaces. Then we'll go
on to talk about operators on a Hilbert space. The hope is to end
the course with the discussion of the spectral theorem for arbitrary
bounded normal operators, although this may end up being covered in
255B this Winter.
Homework assignments:
- Week 1: Read Chapter 1. Do the following problems: p. 6: 4, 6, 8, 9; p. 11: 1, 2; p. 13: 2,
3, 5, 6. In addition, find an R-linear
inclusion i of R2 into C2, which is isometric
(i.e., ||i(h)|| = ||h|| for all h) but which does not preserve
inner products. Why does this not contradict the fact that the norm
||·|| determines the inner product?
- Week 2: Read Chapter 2. Do the following problems: p. 30: 4, 6, 8; p. 36: 11, 12, 15; p. 40:
4, 6, 9, 11; p. 45: 4. And for extra credit: Let T be a bounded operator on a
Hilbert space H. Does
there exists a non-trivial invariant subspace (i.e., a closed subspace K in H, for which TK is a subset of K)?
- Week 3: Read Chapter 3. Do the following problems:
p. 67: 3, 5, 8; p. 70: 1; p. 77: 2; p. 81: 5, 8, 9.
- Week 4: Read Chapter 4. Do the following problems:
p. 95: 2; p. 97: 1, 2, 5, 6, 10; p. 104: 12, 21.
- Week 5-7: Read Chapter 5. Do the following problems:
p. 128: 6, 7, 8 (in #8, show that hn converges to
h iff ||hn|| converges to ||h||), 10; p.
131: 3.
- Week 8: Read Chapter 7. Do the following problems:
p. 158: 7, 8; p. 194: 1, 2, 3, 5; p. 213: 3, 7.