J. H. Wells, L. R. Williams

Embeddings and Extensions in Analysis

• Broschiertes Buch

Produktbeschreibung

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].

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Produktdetails

• Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge .84
• Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
• Artikelnr. des Verlages: 978-3-642-66039-9
• Softcover reprint of the original 1st ed. 1975
• Seitenzahl: 124
• Erscheinungstermin: 11. November 2011
• Englisch
• Abmessung: 244mm x 170mm x 8mm
• Gewicht: 228g
• ISBN-13: 9783642660399
• ISBN-10: 3642660398
• Artikelnr.: 36114351

Inhaltsangabe

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.