J. H. Wells, L. R. Williams

Embeddings and Extensions in Analysis (eBook, PDF)

• Format: PDF

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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Dieser Download kann aus rechtlichen Gründen nur mit Rechnungsadresse in A, B, BG, CY, CZ, D, DK, EW, E, FIN, F, GR, HR, H, IRL, I, LT, L, LR, M, NL, PL, P, R, S, SLO, SK ausgeliefert werden.

Produktdetails

• Verlag: Springer Berlin Heidelberg
• Seitenzahl: 110
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9783642660375
• Artikelnr.: 53101662

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.