巴拿赫空间理论引论

Since the study of normed spaces for their own sake evolved rather than arose fully formed, there is some room to disagree about who founded the field. Albert Bennett came close to giving the definition of a normed space in a 1916 paper [23] on an extension of Newton's method for finding roots, and in 1918 Frederic Riesz [195] based a generalization of the Fredholm theory of integral equations on the defining axioms of a complete normed space, though he did not use these axioms to study the general theory of such spaces. According to Jean Dieudonne [64], Riesz had at this time considered developing a general theory of complete normed spaces, but never published anything in this direction. In a paper that appeared in 1921, Eduard Helly [102] proved what is now called Helly's theorem for bounded linear functionals. Along the way, he developed some of the general theory of normed spaces, but only in the context of norms on subspaces of the vector space of all sequences of complex scalars. 本书为英文版。

Preface 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 Bake 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 Jamess Weak Compacteness Theorem 2.10 Support Points and Subreflexivity 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 Jamess Space J 5 Rotundity and Smoothness 5.1 Rotundity 5.2 Uniform Rotundity 5.3 Generalztions of Uniform Rotundity 5.4 Smoothness 5.5 Unifrom Smoothness 5.6 Generaliztions of Unifrom Smoothness A Prerequistes B Metric Spaces C The Spaces lp and en/p， 1≤p≤∞ D Ultranets References List of Symbols Index