Math 395, Fall 2010.
Honors Analysis I
My office is 4859 East Hall,
and my office hours are Tuesdays 2-5pm and by appointment.
The class meets on Mondays, Tuesdays, Wednesdays and Fridays, 1pm-2pm.
The MWF classes meet in room 130, and the Tuesday
classes meet in room 337, of the David M.
Dennison Building.
The course assistant is David Montague, continuing from last year.
His office hours are on Thursdays, 5-6pm (exception: 6-7pm on September
16), in 3866 East Hall.
His review sessions are on Mondays, 5-6pm, in 3866 East Hall.
Syllabus in PDF
Final grade policies in PDF
The take-home midterm is scheduled for October 22-25.
The take-home final is scheduled for December 10-13.
Exams
Take-home midterm in PDF
Midterm solutions in PDF
Take-home final exam in PDF
Final exam solutions in PDF
Homeworks
Homework 1 (due Friday, September 17, at the beginning of class)
in PDF
Solutions to Homework 1 in PDF
Homework 2 (due Friday, September 24, at the beginning of class) in PDF
Solutions to Homework 2 in PDF
Homework 3 (due Friday, October 1, at the beginning of class) in PDF
Solutions to Homework 3 in PDF
Homework 4 (due Friday, October 15, at the
beginning of class) in PDF
Solutions to Homework 4 in PDF
Extra credit problem set in PDF
Homework 5 (due Friday, October 29, at the
beginning of class) in PDF
Solutions to Homework 5 in PDF
Homework 6 (due Friday, November 5, at the
beginning of class) in PDF
Solutions to Homework 6 in PDF
Homework 7 (due Friday,
November 12, at the
beginning of class) in PDF
Solutions to Homework 7 in PDF
Homework 8 (due Friday,
November 19, at the
beginning of class) in PDF
Solutions to Homework 8 in PDF
Homework 9 (due Friday,
December 3, at the
beginning of class) in PDF
Solutions to Homework 9 in PDF
Homework 10 (due Friday,
December 10, at the
beginning of class) in PDF
Solutions to Homework 10 in PDF
Lecture
summaries (arranged by week)
Preliminaries in PDF
- Introduction to measure theory and integration (week of
September 7)
- Notes on measure spaces in PDF
- Definition and basic properties of integrals (week of
September 13)
- Definition of the integral in PDF
- Basic properties of integrals ( PDF )
- Limit theorems in integration theory ( PDF )
- Product measures and Fubini's theorem (week of
September 20)
- Introduction to Fubini's theorem and Dynkin's lemma
( PDF )
- Statement of Fubini's theorem ( PDF )
- Proof of Fubini's theorem ( PDF )
- More on Fubini's theorem (week of September 27)
- Most of this week was devoted to explaining the
details of
the proof of Fubini's theorem in class.
- No notes for this week will be posted.
- Construction and main properties of the Lebesgue
measure (week of October 4)
- Construction of the Lebesgue measure ( PDF )
- Introduction to topological spaces and metric spaces
(week of October 11)
- Notes on topological and metric spaces in PDF
- Some counterexamples in topology in PDF
- Regularity properties of measures (week of October 20
and 22)
- Notes on regularity of measures in PDF
- Integral inequalities and L^p spaces (week of October
25)
- Statements of Jensen's, Hölder's
and Minkowski's inequalities in PDF
- Proof of Minkowski's inequality in PDF
- Definition of L^p spaces and proof of completeness
in PDF
- Introduction to Fourier Analysis (week of November 1)
- Part of the week was devoted to the definition of
Fourier transforms
and proving its basic properties (Riemann-Lebesgue Lemma).
- We also started discussing orthonormal bases in
Hilbert spaces.
- Notes on real and complex measures in PDF (these notes are not
directly related to Fourier analysis, but Theorem 15.1 in the notes
-- the Radon-Nikodym theorem for complex measures -- is very
useful for two of this week's homework problems.
- Fourier series: definitions and main results (week of
November 8)
- Notes on the structure of Hilbert spaces in PDF
- Introduction to Fourier series in PDF
- Applications of approximation techniques (week of
November 15)
- Introduction to Fourier transforms in PDF
- Notes on convolutions and applications in PDF
- Proofs of some fundamental results on Fourier
transforms (week of November 22)
- Notes on Shwartz functions, Fourier inversion and
Fourier transforms for L^2 in PDF
- Happy Thanksgiving!
- More proofs of theorems about Fourier transforms (week
of November 29)
- The notes for this week are contained in the
previous link
- Contour integration and calculation of Fourier
transforms (week of December 6)
- Notes on contour integration in PDF