Linear Functional Analysis

Imperial College, University of London, 2003. - 122 pages.
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.
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.