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%.
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)
Weierstrass Approximation theorem. Stone-Weierstrass theorem. Baire's theorem.
Baire's theorem. Semi-norms. Convex balanced absorbing sets. The Minkowski functional.
Locally convex topological vector spaces. Characterization through sufficient families of seminorms.
Locally convex topology on smooth functions. Convolution.
Smoothing by convolution. Approximation of compactly supported continuous functions by compactly supported smooth functions. A topology on compactly supported smooth functions.
Norms. Quasi-norms. L^p norms.
Total variation of a signed measure. Jordan's decomposition. Hahn decomposition.
Pre-Hilbert spaces. Inner products. Complete topological vector spaces (Banach, Frechet and Hilbert spaces).
Completeness of L^p. Continuous linear operators. Bornologic spaces.
Distributions. Distributional derivatives.
Sobolev's spaces. Completeness of Sobolev's spaces. The completion of a quasi-normed space.
Density of smooth functions in Sobolev space for R^n. Factor space of a Banach space.
Partition of unity.
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.
Rapidly decreasing functions. Schwartz space. Fourier transform of rapidly decreasing functions. Fourier inversion formula.
Parseval's relation. Poisson summation formula. Tempered distributions.
Fourier transform of tempered distributions. Riesz representation theorem (for Hilbert spaces). Plancherel's theorem.
Compact operators. Integral operators. Integral operators on compact spaces are compact.
Hilbert-Schmidt operators (compactness of). Integral operators with L^2 kernels are Hilbert-Schmidt. Spectral theorem for self-adjoint compact operators.
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.
Topologies on operator spaces: uniform, weak and strong.
Topological groups. Representations of topological groups.
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.
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.
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.
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.
The Krein-Millman theorem. Existence of invariant ergodic measures for abelian group actions on compact metric spaces.
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.