Local Theory of Normed Spaces and Its Applications to Convexity

摘要

The local theory of Banach spaces deals with convex bodies in R(sup n) where n isfinite but large. The main theme of the theory is a quantitative study of the structure of such sets and asymptotic estimates of various parameters associated with them as n -> infinity. The theory grew out of, and in a sense can still be considered as a central part of, functional analysis. The main idea in functional analysis is to consider complex elements (say a function or the state of a physical system) as points in a linear space and to investigate the relation between such objects in analogy to the study of points, lines, planes, etc., in geometry using also an appropriate norm for quantitative estimates. The approach leads naturally in many examples to the study of infinite-dimensional Banach spaces, i.e. geometrically to the study of convex sets in infinite dimensional spaces. The definition of convexity involves only 2-dimensional subspaces of a linear space. Many fundamental facts concerning convex sets in the plane or 3-space generalize in a neat and very useful way to infinite dimensions (e.g. the Hahn-Banach or Krein-Milman theorems). These results are responsible for the great initial success of functional analysis and continue to yield new interesting applications to the present.