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]
CONTENT
I. Isometric Embedding -- ยง1. Introduction -- ยง2. Isometric Embedding in Hilbert Space -- ยง3. Functions of Negative Type -- ยง4. Radial Positive Definite Functions -- ยง5. A Characterization of Subspaces of Lp, 1 ? p ? 2 -- II. The Classes N(X) and RPD(X): Integral Representations -- ยง 6. Radial Positive Definite Functions on ?n -- ยง7. Positive Definite Functions on Infinite-Dimensional Linear Spaces -- ยง 8. Functions of Negative Type on Lp Spaces -- ยง9. Functions of Negative Type on ?N -- III. The Extension Problem for Contractions and Isometries -- ยง10. Formulation -- ยง11. The Kirszbraun Intersection Property -- ยง12. Extension of Contractions for Pairs of Banach Spaces -- ยง13. Special Extension Problems -- IV. Interpolation and Lp Inequalities -- ยง14. A Multi-Component Riesz-Thorin Theorem -- ยง15. Lp Inequalities -- ยง16. A Packing Problem in Lp -- V. The Extension Problem for Lipschitz-Hรถlder Maps between Lp Spaces -- ยง17. K-Functions and an Extension Procedure for Bilinear Forms -- ยง18. Examples of K-Functions -- ยง19. The Contraction Extension Problem for the Pairs (L?q,Lp) -- Author Index -- List of Symbols
