Book:Marián Fabian/Functional Analysis and Infinite-Dimensional Geometry

Subject Matter

 * Functional Analysis

Contents
Preface

1 Basic Concepts in Banach Spaces
 * Hölder and Minkowski inequalities, classical spaces $C \closedint 0 1$, $\ell_p$, $c_0$, $L_p \closedint 0 1$
 * Operators, quotient spaces, finite-dimensional spaces, Riesz's lemma, separability
 * Hilbert spaces, orthonormal bases, $\ell_2$
 * Exercises

2 Hahn-Banach and Banach Open Mapping Theorems
 * Hahn-Banach extension and separation theorems
 * Duals of classical spaces
 * Banach open mapping theorem, closed graph theorem, dual operators
 * Exercises

3 Weak Topologies
 * Weak and weak star topology, Banach-Steinhaus uniform boundedness principle, Alaoglu's and Goldstine's theorem, reflexivity
 * Extreme points, Krein-Milman theorem, James boundary, Ekeland's variational principle, Bishop-Phelps theorem
 * Exercises

4 Locally Convex Spaces
 * Local bases, bounded sets, metrizability and normability, finite-dimensional spaces, distributions
 * Bipolar theorem, Mackey topology
 * Carathéodory and Choquet representation; Banach-Dieudonné, Eberlein-Šmulian theorem, Kaplansky theorems, and Banach-Stone theorem
 * Exercises

5 Structure of Banach Spaces
 * Projections and complementability, Auerbach bases
 * Separable spaces as subspaces of $C \closedint 0 1$ and quotients of $\ell_1$, Sobczyk's theorem, Schur's property of $\ell_1$
 * Exercises

6 Schauder Bases

7 Compact Operators on Banach Spaces

8 Differentiability of Norms

9 Uniform Convexity

10 Smoothness and Structure

11 Weakly Compactly Generated Spaces

12 Topics in Weak Topology

References

Index