Functional Analysis

January-April 2008

(taught by Amritanshu Prasad)


Homework: 40% (due before the first lecture of every week).
Homework scores will be based on four problems chosen at random from those assigned every week. Late homework will not be corrected.
Midsem exam: 20% (25th February, 2008; 9:30am-12:30pm).
Final exam: 40% (24th April, 2008; 9:30am-12:30pm).
Grades: A=90-100%; B=80-89%; C=70-79%; D=60-69%; F=0-59%.

Recommended texts

Introductory Real Analysis, by A. N. Kolmogorov and S. V. Fomin.
Real and Complex Analysis, by Walter Rudin.
Introduction to Topology and Modern Analysis, by G. F. Simmons.
Functional Analysis, by Kosaku Yosida.
Essential Results of Functional Analysis, by Robert J. Zimmer.

Assignments (listed by due date)

January 14
January 21
January 28
February 4
February 11
February 18
March 10
March 17
March 24
March 31
April 7
April 14
April 21
Last assignment

Topics covered

7th January: Weierstrass Approximation theorem. Stone-Weierstrass theorem. Baire's theorem.
10th January: Baire's theorem. Semi-norms. Convex balanced absorbing sets. The Minkowski functional.
14th January: Locally convex topological vector spaces. Characterization through sufficient families of seminorms.
16th January: Locally convex topology on smooth functions. Convolution.
21st January: Smoothing by convolution. Approximation of compactly supported continuous functions by compactly supported smooth functions. A topology on compactly supported smooth functions.
24th January: Norms. Quasi-norms. L^p norms.
28th January: Total variation of a signed measure. Jordan's decomposition. Hahn decomposition.
31st January: Pre-Hilbert spaces. Inner products. Complete topological vector spaces (Banach, Frechet and Hilbert spaces).
4th February: Completeness of L^p. Continuous linear operators. Bornologic spaces.
7th February: Distributions. Distributional derivatives.
12th February: Sobolev's spaces. Completeness of Sobolev's spaces. The completion of a quasi-normed space.
14th February: Density of smooth functions in Sobolev space for R^n. Factor space of a Banach space.
19th February: Partition of unity.
25th February: Compactly supported distributions as linear functionals on the space of smooth functions. Substitution of variables in a distribution. Homogeneous distributions. Distributions invariant under a group action.
10th March: Rapidly decreasing functions. Schwartz space. Fourier transform of rapidly decreasing functions. Fourier inversion formula.
13th March: Parseval's relation. Poisson summation formula. Tempered distributions.
18th March: Fourier transform of tempered distributions. Riesz representation theorem (for Hilbert spaces). Plancherel's theorem.
20th March: Compact operators. Integral operators. Integral operators on compact spaces are compact.
26th March: Hilbert-Schmidt operators (compactness of). Integral operators with L^2 kernels are Hilbert-Schmidt. Spectral theorem for self-adjoint compact operators.
27th March: Spectral theorem for commuting families of compact self-adjoint operators. Normal operators. Spectral theorem for compact normal operators. Spectral theorem for commuting families of compact normal operators.
31st March: Topologies on operator spaces: uniform, weak and strong.
3rd April: Topological groups. Representations of topological groups.
7th April: Strong continuity of representations by translation operators on compactly supported continuous functions. Lusin's theorem. Density of compactly supported continuous functions in L^p of a finite measure space. Strong continuity of representations by translation operators on L^p of a finite measure space.
11th April: Weak and weak* topologies. The Hahn-Banach theorem. The Banach-Alaoglu theorem. Compactness of the space of Probability measures on a compact metric space in the weak* topology.
16th April: The Kakutani-Markov fixed point theorem. Fixed point theorem for the action of a compact group on a closed convex set in the unit ball of the dual of a Banach space with weak* topology.
17th April: Von Neumann's theorem on the existence of an invariant probability measure on a compact space on which a compact group operates. Translation-invariant probability measures for compact groups; bi-invariance and uniqueness of such measures. The Peter-Weyl theorem.
21st April: The Krein-Millman theorem. Existence of invariant ergodic measures for abelian group actions on compact metric spaces.

Further reading

This is a rather random list of material I feel would be fun to read now:
Harmonic Analysis on Phase Space, by Gerald B. Folland.
The Fourier Integral and some of its Applications, by Norbert Wiener.
On the role of the Heisenberg group in harmonic analysis by Roger Howe.