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Math 201B: Analysis

Winter 2007

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Announcement

The Final exam is 4:00pm, Friday March 16th.
There will be a review session in 1060 Bainer from 1:40-3:30p.m. on Thursday, March 15.
Solutions to the Final exam

Here are two previous final exams:

Final from 2000

Final from 2003

Outline of topics covered in final:

- Hilbert spaces (Sec. 6.1–6.3)
- Inner product spaces
- Projection theorem
- Orthogonal direct sums
- Bessel's inequality
- Orthonormal bases
- Parseval's theorem

- Lebesgue integration (Sec. 12.1–12.6)
- Definition of measure and the integral
- Montone and dominated convergence theorems
- Fubini's theorem
- L^p-spaces

- Fourier analysis on the circle (Sec. 7.1–7.4)
- Fourier coefficients in L^1
- Riemann-Lebesgue lemma
- Convolution theorem
- Approximate identities
- Completeness of the Fourier basis in L^2
- Weak derivatives
- Sobolev spaces

- Bounded linear operators on a Hilbert space (Sec. 8.1–8.4, Sec. 8.6)
- Orthogonal projections
- Riesz representation theorem
- Adjoints
- Self-adjoint and unitary operators
- Weak convergence in a Hilbert space

- Spectral theory (Sec. 9.1–9.5)
- The spectrum of a bounded linear operator
- Point, continuous, and residual spectrum
- Compact operators
- Spectral theorem for compact self-adjoint operators

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Instructor

John Hunter

e-mail: `jkhunter@ucdavis.edu`

Phone: (530) 752-3189

Office: MSB 3230

Office hours:M 1:30–3:00 p.m. W 1:00–2:30 p.m.

Lectures: MWF 11:00–11:50 a.m., Physics-Geology 140

Discussion session: Tu 11:00–11:50 a.m., Physics-Geology 140
Midterm: In class, Wednesday Feb 14.

Midterm solutions

Sample midterm problems

Brief solutions to sample midterm problems

Final (Code D): In class, Friday Mar 16, 4:00 p.m.–6:00 p.m., Physics-Geology 140

Important dates:

- Last day to add: Friday Jan 19
- Last day to drop: Wednesday Jan 31
- Academic Holidays: Monday Jan 15, Monday Feb 19
- Last day of instruction: Wednesday Mar 14

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TAs

Hillel Raz

e-mail: `hraz@math.ucdavis.edu`

Office: MSB 2125

Office hours: Tu 4:30–6:00 p.m.
Zhihua Zhang

e-mail: `zhangzh@math.ucdavis.edu`

Office: MSB 2145

Office hours: Th 1:10 p.m.–3:00 p.m.

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Grade

The course grade will be based on the Homework, Midterm, and Final scores,
weighted as follows:
- Homework: 50%
- Midterm: 20%
- Final: 30%

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Text

The text for Math 201 is *Applied Analysis*,
by John Hunter and Bruno Nachtergaele, World Scientific, 2001.
(Corrected reprinting, 2005.)
The course this quarter will cover roughly Chapters 6–10 of the text.

You can buy a copy if you wish, or use the online copy of the book
available at these links in
postscript or
pdf formats. Either way is fine.

We welcome your comments on the text. Please send them to
`jkhunter@ucdavis.edu`
or
`bxn@math.ucdavis.edu`.

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Supplementary Notes

Unordered sums

Measure theory

The heat equation

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Homework Assignments

Set 1 (due in class, Friday Jan 12)

Solutions to Set 1

Set 2 (due in class, Friday Jan 19)

Solutions to Set 2

Set 3 (due in class, Friday Jan 26)

Solutions to Set 3

Set 4 (due in class, Friday Feb 2)

Solutions to Set 4

Set 5 (due in class, Monday Feb 12)

Solutions to Set 5

Set 6 (due in class, Friday Feb 23)

Solutions to Set 6

Set 7 (due in class, Friday Mar 2)

Solutions to Set 7

Set 8 (due in class, Friday Mar 9)

Solutions to Set 8

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