LECTURE NOTES OF WILLIAM CHEN
LINEAR FUNCTIONAL ANALYSIS
This set of notes has been organized in such a way to create a single volume suitable for an introduction to some of the basic ideas in linear functional analysis as well as the role of linearity in analysis. Chapters 1 and 2 were used in various forms and on many occasions between 1983 and 1990 by the author at Imperial College, University of London. Chapters 3 - 9 were added in Sydney in 2001.
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Chapter 1 : INTRODUCTION TO METRIC SPACES >>
- Introduction
- Convergence in a Metric Space
- Open Sets and Closed Sets
- Limits and Continuity
Chapter 2 : CONNECTEDNESS, COMPLETENESS AND COMPACTNESS >>
- Connected Metric Spaces
- Complete Metric Spaces
- Compact Metric Spaces
- Continuous Functions with Compact Domains
Chapter 3 : NORMED VECTOR SPACES >>
- Review of Vector Spaces
- Norm in a Vector Space
- Continuity Properties
- Finite Dimensional Normed Vector Spaces
- Linear Subspaces of Normed Vector Spaces
- Banach Spaces
Chapter 4 : INNER PRODUCT SPACES >>
- Introduction
- Inner Product Spaces
- Norm in an Inner Product Space
- Hilbert Spaces
- The Closest Point Property
Chapter 5 : ORTHOGONAL EXPANSIONS >>
- Orthogonal and Orthonormal Systems
- Convergence of Fourier Series
- Orthonormal Bases
- Separable Hilbert Spaces
- Splitting up a Hilbert Space
Chapter 6 : LINEAR FUNCTIONALS >>
- Introduction
- Dual Spaces
- Self Duality of Hilbert Spaces
Chapter 7 : INTRODUCTION TO LINEAR TRANSFORMATIONS >>
- Introduction
- Space of Linear Transformations
- Composition of Linear Transformations
Chapter 8 : LINEAR TRANSFORMATIONS ON HILBERT SPACES >>
- Adjoint Transformations
- Hermitian Operators
Chapter 9 : SPECTRUM OF A LINEAR OPERATOR >>
- Introduction
- Compact Operators