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| Preface | |
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| The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions | |
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| The Analytic Form of the Hahn-Banach Theorem: Extension of Linear Functional | |
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| The Geometric Forms of the Hahn-Banach Theorem: Separation of Convex Sets | |
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| The Bidual E<sup>**</sup>. Orthogonality Relations | |
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| A Quick Introduction to the Theory of Conjugate Convex Functions | |
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| The Uniform Boundedness Principle and the Closed Graph Theorem | |
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| The Baire Category Theorem | |
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| The Uniform Boundedness Principle | |
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| The Open Mapping Theorem and the Closed Graph Theorem | |
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| Complementary Subspaces. Right and Left Invertibility of Linear Operators | |
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| Orthogonality Revisited | |
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| An Introduction to Unbounded Linear Operators. Definition of the Adjoint | |
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| A Characterization of Operators with Closed Range. A Characterization of Surjective Operators | |
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| Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity | |
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| The Coarsest Topology for Which a Collection of Maps Becomes Continuous | |
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| Definition and Elementary Properties of the Weak Topology �(E, E<sup>*</sup>) | |
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| Weak Topology, Convex Sets, and Linear Operators | |
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| The Weak<sup>*</sup> Topology �(E<sup>*</sup> E) | |
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| Reflexive Spaces | |
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| Separable Spaces | |
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| Uniformly Convex Spaces | |
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| L<sup>p</sup> Spaces | |
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| Some Results about Integration That Everyone Must Know | |
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| Definition and Elementary Properties of L<sup>p</sup> Spaces | |
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| Reflexivity. Separability. Dual of L<sup>p</sup> | |
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| Convolution and regularization | |
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| Criterion for Strong Compactness in L<sup>p</sup> | |
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| Hilbert Spaces | |
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| Definitions and Elementary Properties. Projection onto a Closed Convex Set | |
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| The Dual Space of a Hilbert Space | |
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| The Theorems of Stampacchia and Lax-Milgram | |
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| Hilbert Sums. Orthonormal Bases | |
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| Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators | |
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| Definitions. Elementary Properties. Adjoint | |
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| The Riesz-Fredholm Theory | |
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| The Spectrum of a Compact Operator | |
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| Spectral Decomposition of Self-Adjoint Compact Operators | |
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| The Hille-Yosida Theorem | |
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| Definition and Elementary Properties of Maximal Monotone Operators | |
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| Solution of the Evolution Problem du/dt + Au = 0 on (0, +∞), u(0) = u<sub>0</sub>. Existence and uniqueness | |
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| Regularity | |
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| The Self-Adjoint Case | |
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| Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension | |
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| Motivation | |
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| The Sobolev Space W<sup>1.p</sup> (I) | |
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| The Space &$$$; | |
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| Some Examples of Boundary Value Problems | |
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| The Maximum Principle | |
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| Eigenfunctions and Spectral Decomposition | |
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| Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions | |
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| Definition and Elementary Properties of the Sobolev Spaces W<sup>1.p</sup> (�) | |
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| Extension Operators | |
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| Sobolev Inequalities | |
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| The Space &$$$; (�) | |
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| Variational Formulation of Some Boundary Value Problems | |
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| Regularity of Weak Solutions | |
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| The Maximum Principle | |
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| Eigenfunctions and Spectral Decomposition | |
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| Evolution Problems: The Heat Equation and the Wave Equation | |
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| The Heat Equation: Existence, Uniqueness, and Regularity | |
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| The Maximum Principle | |
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| The Wave Equation | |
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| Miscellaneous Complements | |
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| Finite-Dimensional and Finite-Codimensional Spaces | |
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| Quotient Spaces | |
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| Some Classical Spaces of Sequences | |
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| Banach Spaces over C: What Is Similar and What Is Different? | |
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| Solutions of Some Exercises | |
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| Problems | |
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| Partial Solutions of the Problems | |
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| Notation | |
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| References | |
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| Index | |