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
 * Shrinking and boundedly complete bases, reflexivity, Mazur’s basic sequence theorem, small perturbation lemma
 * Bases in classical spaces: block basis sequences, Pełczyński decomposition method and subspaces of $\ell_p$, Pitt’s theorem, Khintchine’s inequality and subspaces of $L_p$
 * Unconditional bases, James’s theorem on containment of $\ell_1$ and $c_0$, James's space $J$, Bessaga-Pełczyński theorem
 * Markushevich bases: existence for separable spaces, extension property, Johnson’s and Plichko’s result on $\ell_\infty$
 * Exercises

7 Compact Operators on Banach Spaces
 * Compact operators and finite rank operators, Fredholm operators, Fredholm alternative
 * Spectral theory: eigenvalues, spectrum, resolvent, eigenspaces
 * Self-adjoint operators, spectral theory of compact self-adjoint and compact normal operators
 * Fixed points: Banach’s contraction principle, non-expansive mappings, Ryll-Nardzewski theorem, Brouwer’s and Schauder’s theorems, invariant subspaces
 * Exercises

8 Differentiability of Norms
 * Šmulian’s dual test, Kadec’s Fréchet-smooth renorming of spaces with separable dual, Fréchet differentiability of convex functions
 * Extremal structure, Lindenstrauss’ result on strongly exposed points and norm attaining operators
 * Exercises

9 Uniform Convexity
 * Uniform convexity and uniform smoothness, $\ell_p$ spaces
 * Finite representability, local reflexivity, superreflexive spaces and Enflo's renorming, Kadec’s and Gurarii-Gurarii-James theorems
 * Exercises

10 Smoothness and Structure
 * Variational principles (smooth and compact), subdifferential, Stegall’s variational principle
 * Smooth approximation: partitions of unity
 * Lipschitz homeomorphisms, Aharoni’s embeddings into $c_0$, Heinrich-Mankiewicz results on linearization of Lipschitz maps
 * Homeomorphisms: Mazur’s theorem on $\ell_p$, Kadec’s theorem
 * Smoothness in $\ell_p$, Hilbert spaces
 * Countable James boundary and saturation by $c_0$
 * Exercises

11 Weakly Compactly Generated Spaces
 * Projectional resolutions, injections into $\map {c_0} \Gamma$, Eberlein compacts, embedding into a reflexive space, locally uniformly rotund and smooth renormings
 * Weakly compact operators, Davis-Figiel-Johnson-Pełczyński factorization, absolutely summing operators, Pietsch factorization, Dunford-Pettis property
 * Quasicomplements
 * Exercises

12 Topics in Weak Topology
 * Eberlein compacts, metrizable subspaces
 * Uniform Eberlein compacts, scattered compacts
 * Weakly Lindelöf spaces, property C
 * Corson compacts, weak pseudocompactness in Banach spaces, $\struct {B_X, w}$ Polish
 * Exercises

References

Index