Metrization theorem

Abstract: 1 Basic Concepts.- 1.1 Preliminaries.- 1.2 Norms.- 1.3 First Properties of Normed Spaces.- 1.4 Linear Operators Between Normed Spaces.- 1.5 Baire Category.- 1.6 Three Fundamental Theorems.- 1.7 Quotient Spaces.- 1.8 Direct Sums.- 1.9 The Hahn-Banach Extension Theorems.- 1.10 Dual Spaces.- 1.11 The Second Dual and Reflexivity.- 1.12 Separability.- 1.13 Characterizations of Reflexivity.- 2 The Weak and Weak Topologies.- 2.1 Topology and Nets.- 2.2 Vector Topologies.- 2.3 Metrizable Vector Topologies.- 2.4 Topologies Induced by Families of Functions.- 2.5 The Weak Topology.- 2.6 The Weak Topology.- 2.7 The Bounded Weak Topology.- 2.8 Weak Compactness.- 2.9 James's Weak Compactness Theorem.- 2.10 Extreme Points.- 2.11 Support Points and Subreflexivity.- 3 Linear Operators.- 3.1 Adjoint Operators.- 3.2 Projections and Complemented Subspaces.- 3.3 Banach Algebras and Spectra.- 3.4 Compact Operators.- 3.5 Weakly Compact Operators.- 4 Schauder Bases.- 4.1 First Properties of Schauder Bases.- 4.2 Unconditional Bases.- 4.3 Equivalent Bases.- 4.4 Bases and Duality.- 4.5 James's Space J.- 5 Rotundity and Smoothness.- 5.1 Rotundity.- 5.2 Uniform Rotundity.- 5.3 Generalizations of Uniform Rotundity.- 5.4 Smoothness.- 5.5 Uniform Smoothness.- 5.6 Generalizations of Uniform Smoothness.- A Prerequisites.- B Metric Spaces.- D Ultranets.- References.- List of Symbols.

Abstract: Preface.- Introduction. The Krein-Milman theorem as an integral representation theorem.- Application of the Krein-Milman theorem to completely monotonic functions.- Choquet's theorem: The metrizable case.- The Choquet-Bishop-de Leeuw existence theorem.- Applications to Rainwater's and Haydon's theorems.- A new setting: The Choquet boundary.- Applications of the Choquet boundary to resolvents.- The Choquet boundary for uniform algebras.- The Choquet boundary and approximation theory.- Uniqueness of representing measures.- Properties of the resultant map.- Application to invariant and ergodic measures.- A method for extending the representation theorems: Caps.- A different method for extending the representation theorems.- Orderings and dilations of measures.- Additional Topics.- References.- Index of symbols.- Index.

Abstract: Publisher Summary This chapter discusses integers and topology. Role in topology of certain cardinals is associated with ω. This chapter discusses the problems on first countability, convergence, and separable metrizable spaces. A typical use of these set theoretic cardinals associated with ω involves topologically defined cardinals. Another use of these set theoretic cardinals associated with ω is that certain topological results hold if one of these cardinals equals ω1. The chapter also discusses the set theory, which states an ordinal is the set of smaller ordinals, and a cardinal is an initial ordinal. ω is ω0, and c is 2ω. The chapter also describes sequential and countable compactness. A countable set A of a space X is said to cluster at x ∈ X if each neighborhood of x contains infinitely many points of A, and it is said to converge to x ∈ X if each neighborhood of x contains all but finitely many points of A. A space is called countably compact if each countably infinite set clusters at some point, and it is called sequentially compact if each countably infinite set has an infinite subset that converges somewhere. Moreover, a space X is called subsequential if for every countably infinite A ⊆ X and for every cluster point x of A, there is an infinite subset of A that converges to x.

Abstract: 1. Topological spaces, normality and compactness 2. Paracompact and pseudo- metrizable spaces 3. Covering dimension 4. Inductive dimension 5. Local dimension 6. Images of zero-dimensional spaces 7. The dimension of pseudo-metrizable and metrizable spaces 8. The dimension of bicompact spaces.