Linear Functional Analysis

Description

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.