LECTURE NOTES OF WILLIAM CHEN
# LINEAR FUNCTIONAL ANALYSIS

### Chapter 1 : INTRODUCTION TO METRIC SPACES >>

### Chapter 2 : CONNECTEDNESS, COMPLETENESS AND COMPACTNESS >>

### Chapter 3 : NORMED VECTOR SPACES >>

### Chapter 4 : INNER PRODUCT SPACES >>

### Chapter 5 : ORTHOGONAL EXPANSIONS >>

### Chapter 6 : LINEAR FUNCTIONALS >>

### Chapter 7 : INTRODUCTION TO LINEAR TRANSFORMATIONS >>

### Chapter 8 : LINEAR TRANSFORMATIONS ON HILBERT SPACES >>

### Chapter 9 : SPECTRUM OF A LINEAR OPERATOR >>

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.

To read the notes, click the links below for connection to the appropriate PDF files.

The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.

- Introduction
- Convergence in a Metric Space
- Open Sets and Closed Sets
- Limits and Continuity

- Connected Metric Spaces
- Complete Metric Spaces
- Compact Metric Spaces
- Continuous Functions with Compact Domains

- Review of Vector Spaces
- Norm in a Vector Space
- Continuity Properties
- Finite Dimensional Normed Vector Spaces
- Linear Subspaces of Normed Vector Spaces
- Banach Spaces

- Introduction
- Inner Product Spaces
- Norm in an Inner Product Space
- Hilbert Spaces
- The Closest Point Property

- Orthogonal and Orthonormal Systems
- Convergence of Fourier Series
- Orthonormal Bases
- Separable Hilbert Spaces
- Splitting up a Hilbert Space

- Introduction
- Dual Spaces
- Self Duality of Hilbert Spaces

- Introduction
- Space of Linear Transformations
- Composition of Linear Transformations

- Adjoint Transformations
- Hermitian Operators

- Introduction
- Compact Operators