Notes on Functional Analysis

These notes are a record of a one semester course on Functional Analysis given by the author to second year Master of Statistics students at the Indian Statistical Institute, New Delhi. Students taking this course have a strong background in real analysis, linear algebra, measure theory and probability, and the course proceeds rapidly from the definition of a normed linear space to the spectral theorem for bounded selfadjoint operators in a Hilbert space.

The book is organised as twenty six lectures, each corresponding to a ninety minute class session. This may be helpful to teachers planning a course on this topic. Well prepared students can read it on their own.

Contents

1: Banach Spaces
2: Dimensionality
3: New Banach Spaces from Old
4: The Hahn-Banach Theorem
5: The Uniform Boundedness Principle
6: The Open Mapping Theorem
7: Dual Spaces
8: Some Applications
9: The Weak Topology
10: The Second Dual and the Weak* Topology
11: Hilbert Spaces
12: Orthonormal Bases
13: Linear Operators
14: Adjoint Operators
15: Some Special Operators in Hilbert Space
16: The Resolvent and The Spectrum
17: Subdivision of the Spectrum
18: Spectra of Normal Operators
19: Square Roots and the Polar Decomposition
20: Compact Operators
21: The Spectrum of a Compact Operator
22: Compact Operators and Invariant Subspaces
23: Trace Ideals
24: The Spectral Theorem –I
25: The Spectral Theorem –II
26: The Spectral Theorem –III
Index