Book
Graduate Studies in Mathematics, Volume XXX,
Amer. Math. Soc., Providence, (to appear).
Topics in Linear and Nonlinear Functional Analysis
Gerald Teschl
Abstract
This manuscript provides a brief introduction to linear and nonlinear Functional Analysis.
There is also an accompanying text on Real Analysis.
MSC: 4601, 46E30, 47H10, 47H11, 58Exx, 76D05
Keywords: Functional Analysis, Banach space, Hilbert space, Mapping degree, fixedpoint theorems,
differential equations, NavierStokes equation
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Table of contents

Preface
 A first look at Banach and Hilbert spaces
 Introduction: Linear partial differential equations
 The Banach space of continuous functions
 The geometry of Hilbert spaces
 Completeness
 Compacteness
 Bounded operators
 Sums and quotients of Banach spaces
 Spaces of continuous and differentiable functions
 Hilbert spaces
 Hilbert spaces
 The projection theorem and the Riesz lemma
 Operators defined via forms
 Orthogonal sums and tensor products
 Applications to Fourier series
 Compact operators
 Compact operators
 The spectral theorem for compact symmetric operators
 Applications to SturmLiouville operators
 Estimating eigenvalues
 Singular value decomposition of compact operators
 HilbertSchmidt and trace class operators
 The main theorems about Banach spaces
 The Baire theorem and its consequences
 The HahnBanach theorem and its consequences
 Refexivity
 The adjoint operator
 Weak convergence
 Bounded linear operators
 Banach algebras
 The C^{*} algebra of operators and the spectral theorem
 Spectral measures
 More on convexity
 The geometric HahnBanach theorem
 Convex sets and the KreinMilman theorem
 Weak topologies
 Beyond Banach spaces: Locally convex spaces
 Uniformly convex spaces
 Advanced spectral theory
 The Gelfand representation theorem
 Spectral theory for compact operators
 Fredholm operators
 Unbounded linear operators
 Closed operators
 Spectral theory for unbounded operators
 Reducing subspaces and spectral projections
 Relatively bounded and relatively compact operators
 Unbounded Fredholm operators
 Analysis in Banach spaces
 Single variable calculus in Banach spaces
 Multivariable calculus in Banach spaces
 Minimizing nonlinear functionals via calculus
 Minimizing nonlinear functionals II via compactness
 Contraction principles
 Ordinary differential equations
 The Brouwer mapping degree
 Introduction
 Definition of the mapping degree and the determinant formula
 Extension of the determinant formula
 The Brouwer fixed point theorem
 Kakutani's fixed point theorem and applications to game theory
 Further properties and extensions
 The Jordan curve theorem
 The LeraySchauder mapping degree
 The mapping degree on finite dimensional Banach spaces
 Compact operators
 The LeraySchauder mapping degree
 The LeraySchauder principle and the Schauder fixed point theorem
 Applications to integral and differential equations
 Monotone operators
 Monotone operators
 The nonlinear LaxMilgram theorem
 The main theorem of monotone operators
 Appendix: Some set theory
 Appendix: Metric and topological spaces
 Basics
 Convergence and completeness
 Functions
 Product topologies
 Compactness
 Connectedness
 Constructing continuous functions
 Initial and final topologies
 Continuous functions on metric spaces
Glossary of notations
Index