Bhatia R. Notes on Functional Analysis

New Delhi: Hindustan Book Agency, 2009. — 248 p. — (Texts and Readings in Mathematics, 50). — ISBN 8185931895.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.Preface.
A word about notation.
Banach Spaces.
Dimensionality.
New Banach Spaces from Old.
The Hahn-Banach Theorem.
The Uniform Boundedness Principle.
The Open Mapping Theorem.
Dual Spaces.
Some Applications.
The Weak Topology.
The Second Dual and the Weak* Topology.
Hilbert Spaces.
Orthonormal Bases.
Linear Operators.
Adjoint Operators.
Some Special Operators in Hilbert Space.
The Resolvent and The Spectrum.
Subdivision of the Spectrum.
Spectra of Normal Operators.
Square Roots and the Polar Decomposition.
Compact Operators.
The Spectrum of a Compact Operator.
Compact Operators and Invariant Subspace.
Trace Ideals.
The Spectral Theorem-I.
The Spectral Theorem-II.
The Spectral Theorem-III.
Index.True PDF