Linear Functional Analysis

by WWL Chen

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. Chapters 1-2 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 chapters 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.

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