The object of this book is a presentation of the major results relating to two geometrically inspired problems in analysis. One is that of determining which metric spaces can be isometrically embedded in a Hilbert space or, more generally, P in an L space; the other asks for conditions on a pair of metric spaces which will ensure that every contraction or every Lipschitz-Holder map from a subset of X into Y is extendable to a map of the same type from X into Y. The initial work on isometric embedding was begun by K. Menger [1928] with his metric investigations of Euclidean geometries and continued, in its analytical formulation, by I. J. Schoenberg [1935] in a series of papers of classical elegance. The problem of extending Lipschitz-Holder and contraction maps was first treated by E. J. McShane and M. D. Kirszbraun [1934]. Following a period of relative inactivity, attention was again drawn to these two problems by G. Minty's work on non-linear monotone operators in Hilbert space [1962]; by S. Schonbeck's fundamental work in characterizing those pairs (X,Y) of Banach spaces for which extension of contractions is always possible [1966]; and by the generalization of many of Schoenberg's embedding theorems to the P setting of L spaces by Bretagnolle, Dachuna Castelle and Krivine [1966].
Inhalt
I. Isometric Embedding.- Introduction.- Isometric Embedding in Hilbert Space.- Functions of Negative Type.- Radial Positive Definite Functions.- A Characterization of Subspaces of Lp, 1 ? p ? 2.- II. The Classes N(X) and RPD(X): Integral Representations.- 6. Radial Positive Definite Functions on ?n.- Positive Definite Functions on Infinite-Dimensional Linear Spaces.- 8. Functions of Negative Type on Lp Spaces.- Functions of Negative Type on ?N.- III. The Extension Problem for Contractions and Isometries.- . Formulation.- . The Kirszbraun Intersection Property.- . Extension of Contractions for Pairs of Banach Spaces.- . Special Extension Problems.- IV. Interpolation and Lp Inequalities.- . A Multi-Component Riesz-Thorin Theorem.- . Lp Inequalities.- . A Packing Problem in Lp.- V. The Extension Problem for Lipschitz-Hder Maps between Lp Spaces.- . K-Functions and an Extension Procedure for Bilinear Forms.- . Examples of K-Functions.- . The Contraction Extension Problem for the Pairs (L?q,Lp).- Author Index.- List of Symbols.