Embeddings and Extensions in Analysis

by J.H. Wells, L.R. Williams

Overview

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].

Product Details

ISBN-13:	9783642660399
Publisher:	Springer Berlin Heidelberg
Publication date:	11/11/2011
Series:	Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge , #84
Edition description:	Softcover reprint of the original 1st ed. 1975
Pages:	110
Product dimensions:	6.69(w) x 9.61(h) x 0.01(d)

Table of Contents

I. Isometric Embedding.- §1. Introduction.- §2. Isometric Embedding in Hilbert Space.- §3. Functions of Negative Type.- §4. Radial Positive Definite Functions.- §5. 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.- §7. Positive Definite Functions on Infinite-Dimensional Linear Spaces.- § 8. Functions of Negative Type on Lp Spaces.- §9. Functions of Negative Type on—N.- III. The Extension Problem for Contractions and Isometries.- §10. Formulation.- §11. The Kirszbraun Intersection Property.- §12. Extension of Contractions for Pairs of Banach Spaces.- §13. Special Extension Problems.- IV. Interpolation and Lp Inequalities.- §14. A Multi-Component Riesz-Thorin Theorem.- §15. Lp Inequalities.- §16. A Packing Problem in Lp.- V. The Extension Problem for Lipschitz-Hölder Maps between Lp Spaces.- §17. K-Functions and an Extension Procedure for Bilinear Forms.- §18. Examples of K-Functions.- §19. The Contraction Extension Problem for the Pairs (L?q,Lp).- Author Index.- List of Symbols.