Functional Analysis and Infinite-Dimensional Geometry

Cover
Title page
Preface
1 Basic Concepts in Banach Spaces
Hölder and Minkowski inequalities, classical spaces C[0,1], l_p, C₀, L_p[0,1]
Operators, quotient spaces, finite-dimensional spaces, Riesz's lemma, separability
Hilbert spaces, orthonormal bases, l₂
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-Smulian, Kaplansky theorems, and Banach-Stone theorem
Exercises
5 Structure of Banach Spaces
Projections and complementability, Auerbach bases
Separable spaces as subspaces of C[0,1] and quotients of l₁, Sobczyk's theorem, Schur's property of l₁
Exercises
6 Schauder Bases
Shrinking and boundedly complete bases, reflexivity, Mazur's basic sequence theorem, small perturbation lemma
Block basis sequences, Pelczynski's decomposition method and subspaces of l_p, Pitt's theorem, Khintchine's inequality and subspaces of L_p 1
Unconditional bases, James's theorem on containment of l₁ and c₀, James's space J, Bessaga-Pelczynski theorem
Markushevich bases in separable spaces, their extension property, Johnson's and Plichko's result on l_∞
Exercises
7 Compact Operators on Banach Spaces
Compact and finite-rank operators, Fredholm operators, Fredholm alternative
Eigenvalues, eigenspaces, spectrum, spectral decomposition
Spectral theory of compact self-adjoint and compact normal operators
Banach's contraction principle, nonexpansive mappings, Ryll-Nardzewski theorem, Brouwer's and Schauder's theorems, invariant subspaces
Exercises
8 Differentiability of Norms
Smulian's dual test, Kadec's Fréchet-smooth renorming of spaces with separable dual, Fréchet differentiability of convex functions
More on extremal structure, Lindenstrauss's result on strongly exposed points and norm-attaining operators
Exercises
9 Uniform Convexity
Uniform convexity and uniform smoothness, l_p spaces
Finite representability, local reflexivity, superreflexive spaces and Enflo's renorming, Kadec's and Gurarii Gurarii-James theorems
Exercises
10 Smoothness and Structure
Smooth and compact variational principles, subdifferential, Stegall's variational principle
Partitions of unity, smooth approximation
Lipschitz homeomorphisms, Aharoni's embeddings into c₀, Heinrich-Mankiewicz results on linearization of Lipschitz maps
Homeomorphisms, Mazur's theorem on l_p, Kadec's theorem
Smoothness in l_p and Hilbert spaces
Countable James boundary and saturation by c₀
Exercises
11 Weakly Compactly Generated Spaces
Projectional resolutions, injections into c₀(Γ), Eberlein compacts, embedding into a reflexive space, locally uniformly rotund and smooth renormings
Weakly compact operators, Davis-Figiel-Johnson-Pelczynski 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,(B_X,w) Polish
Exercises
References
Index