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" Embeddings and Extensions in Analysis "
by J.H. Wells, L.R. Williams.
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BL
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752265
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b572224
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| Main Entry
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by J.H. Wells, L.R. Williams.
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Embeddings and Extensions in Analysis\ by J.H. Wells, L.R. Williams.
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| Publication Statement
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Berlin, Heidelberg : Springer Berlin Heidelberg, 1975
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| Series Statement
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Ergebnisse der Mathematik und ihrer Grenzgebiete Band 84, A Series of Modern Surveys in Mathematics, 84.
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| ISBN
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3642660371
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: 3642660398
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: 9783642660375
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: 9783642660399
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| Contents
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I. Isometric Embedding --; ʹ1. Introduction --; ʹ2. Isometric Embedding in Hilbert Space --; ʹ3. Functions of Negative Type --; ʹ4. Radial Positive Definite Functions --; ʹ5. A Characterization of Subspaces of Lp, 1? p? 2 --; II. The Classes N(X) and RPD(X): Integral Representations --; ʹ 6. Radial Positive Definite Functions on?n --; ʹ7. Positive Definite Functions on Infinite-Dimensional Linear Spaces --; ʹ 8. Functions of Negative Type on Lp Spaces --; ʹ9. Functions of Negative Type on?N --; III. The Extension Problem for Contractions and Isometries --; ʹ10. Formulation --; ʹ11. The Kirszbraun Intersection Property --; ʹ12. Extension of Contractions for Pairs of Banach Spaces --; ʹ13. Special Extension Problems --; IV. Interpolation and Lp Inequalities --; ʹ14. A Multi-Component Riesz-Thorin Theorem --; ʹ15. Lp Inequalities --; ʹ16. A Packing Problem in Lp --; V. The Extension Problem for Lipschitz-Hölder Maps between Lp Spaces --; ʹ17. K-Functions and an Extension Procedure for Bilinear Forms --; ʹ18. Examples of K-Functions --; ʹ19. The Contraction Extension Problem for the Pairs (L?q, Lp) --; Author Index --; List of Symbols.
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| Abstract
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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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Mathematics.
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J H Wells
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L R Williams
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