Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras

Preface
	Aim of This Book
	Audience
	Acknowledgments
Contents
1 Introduction
	1.1 Preliminaries
Part I Metric Spaces
	2 Distance
		2.1 Balls and Open Sets
			Open Sets
				Relatively Open Sets
		2.2 Closed Sets
			Limit Points and Dense Subsets
	3 Convergence and Continuity
		3.1 Convergence
		3.2 Continuity
			Homeomorphisms
	4 Completeness and Separability
		4.1 Completeness
		4.2 Uniformly Continuous Maps
		4.3 Separable Spaces
	5 Connectedness
		5.1 Connected Sets
		5.2 Components
	6 Compactness
		6.1 Bounded Sets
		6.2 Totally Bounded Sets
		6.3 Compact Sets
		6.4 The Space C(X,Y)
Part II Banach and Hilbert Spaces
	7 Normed Spaces
		7.1 Vector Spaces
		7.2 Norms
		7.3 Metric and Vector Properties
			Connected and Compact Subsets
		7.4 Complete and Separable Normed Vector Spaces
		7.5 Series
			Convergence Tests
	8 Continuous Linear Maps
		8.1 Operators
		8.2 Operator Norms
			Matrix Norms
		8.3 Isomorphisms and Projections
			Projections
		8.4 Quotient Spaces
		8.5 Rn and Totally Bounded Sets
	9 The Classical Spaces
		9.1 Sequence Spaces
			The Space ∞
			The Space 1
			The Space 2
			The Space p
		9.2 Function Spaces
			Lebesgue Measure on Rn
			Measurable Functions
			Integrable Functions
			Integral Operators
			Approximation of Functions
			The Fourier Series
	10 Hilbert Spaces
		10.1 Inner Products
		10.2 Least Squares Approximation
			Least Squares Approximation
		10.3 Duality H*H
			The Adjoint Map T*
		10.4 Inverse Problems
		10.5 Orthonormal Bases
			Fourier Expansion
			Examples of Orthonormal Bases
				L2[a,b]—Fourier Series
				L2[-1,1]—Legendre Polynomials
				L2[0,∞[—Laguerre Functions
				L2(R)—Hermite Functions
			Applications of Orthonormal Bases
	11 Banach Spaces
		11.1 The Open Mapping Theorem
			Complementary Subspaces
		11.2 Compact Operators
			Fredholm Operators
		11.3 The Dual Space X*
			Annihilators
			The Double Dual X**
		11.4 The Adjoint T
		11.5 Strong and Weak Convergence
			Weak Convergence
			Weak Convergence in p
	12 Differentiation and Integration
		12.1 Differentiation
		12.2 Integration for Vector-Valued Functions
			Application: The Newton-Raphson Algorithm
		12.3 Complex Differentiation and Integration
Part III Banach Algebras
	13 Banach Algebras
		13.1 Introduction
			Subalgebras and Ideals
			Morphisms
			Representation in B(X)
		13.2 Power Series
			The Exponential and Logarithm Maps
		13.3 The Group of Invertible Elements
		13.4 Analytic Functions
	14 Spectral Theory
		14.1 The Spectrum of T
			The Spectral Radius
		14.2 The Spectrum of an Operator
			The Spectrum of the Adjoint
		14.3 Spectra of Compact Operators
			Ascents and Descents
			The Spectrum of a Compact Operator
		14.4 The Functional Calculus
		14.5 The Gelfand Transform
			Quasinilpotents and the Radical
			The State Space
			The Gelfand Transform
	15 C*-Algebras
		15.1 Normal Elements
		15.2 Normal Operators in B(H)
		15.3 The Numerical Range
		15.4 The Spectral Theorem for Compact Normal Operators
			The Singular Value Decomposition
			Application: Feature Extraction
		15.5 Ideals of Compact Operators
			Hilbert-Schmidt Operators
		15.6 Representation Theorems
		15.7 Positive Self-Adjoint Elements
			Polar Decomposition
		15.8 Spectral Theorem for Normal Operators
			Embedding in B(H)
Hints to Selected Problems
Glossary of Symbols
Further Reading
	More Advanced Books
	Other References
	Selected Historical Articles
Index