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. You will need Adobe Acrobat Reader Version 4.0 or later.

The material is available free to all individuals, on the understanding that it is not to be used for financial gains, 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 (last uploaded on 2 December 2002)

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

Chapter 2: CONNECTEDNESS, COMPLETENESS AND COMPACTNESS (last uploaded on 2 December 2002)

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

Chapter 3: NORMED VECTOR SPACES (last uploaded on 2 December 2002)

• 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 (last uploaded on 2 December 2002)

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

Chapter 5: ORTHOGONAL EXPANSIONS (last uploaded on 2 December 2002)

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

Chapter 6: LINEAR FUNCTIONALS (last uploaded on 10 November 2003)

• Introduction
• Dual Spaces
• Self Duality of Hilbert Spaces

Chapter 7: INTRODUCTION TO LINEAR TRANSFORMATIONS (last uploaded on 2 December 2002)

• Introduction
• Space of Linear Transformations
• Composition of Linear Transformations

Chapter 8: LINEAR TRANSFORMATIONS ON HILBERT SPACES (last uploaded on 2 December 2002)

• Adjoint Transformations
• Hermitian Operators

Chapter 9: SPECTRUM OF A LINEAR OPERATOR (last uploaded on 2 December 2002)

• Introduction
• Compact Operators