Smooth Analysis in Banach Spaces

This book is about the subject of higher smoothness in separable real Banach spaces. It brings together several angles of view on polynomials, both in finite and infinite setting. Also a rather thorough and systematic view of the more recent results, and the authors work is given. The book revolves...

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Hlavní autori: Hájek, Petr, Johanis, Michal
Médium: E-kniha Kniha
Jazyk:English
Vydavateľské údaje: Germany De Gruyter 2014
De Gruyter, Inc
Vydanie:1
Edícia:De Gruyter Series in Nonlinear Analysis and Applications
Predmet:
Approximation
Banach Space
Banach spaces
MATHEMATICS
MATHEMATICS / Functional Analysis
MATHEMATICS / Mathematical Analysis
MATHEMATICS / Reference
Mathematics. Analysis
Normed linear spaces
Polynomial
Polynomials
Smoothness
Variational Principle
Polynomial
Banach Space
Approximation
Variational Principle
Smoothness
ISBN:3110258994, 9783110258998, 9783110258981, 3110258986, 9783110391992, 3110391996
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Obsah:
  • Intro -- Introduction -- Chapter 1. Fundamental properties of smoothness -- 1. Multilinear mappings and polynomials -- 2. Complexification -- 3. Fréchet smoothness -- 4. Taylor polynomial -- 5. Smoothness classes -- 6. Power series and their convergence -- 7. Complex mappings -- 8. Analytic mappings -- 9. Notes and remarks -- Chapter 2. Basic properties of polynomials on Rn -- 1. Spaces of polynomials on Rn -- 2. Cubature formulae -- 3. Estimates related to Chebyshev polynomials -- 4. Polynomials and L_p-norms on Rn -- 5. Polynomial identities -- 6. Estimates of coefficients of polynomials -- 7. Notes and remarks -- Chapter 3. Weak continuity of polynomials and estimates of coefficients -- 1. Tensor products and spaces of multilinear mappings -- 2. Weak continuity and spaces of polynomials -- 3. Weak continuity and _1 -- 4. (p,q)-summing operators -- 5. Estimates of coefficients of multilinear mappings -- 6. Bohr radius -- 7. Notes and remarks -- Chapter 4. Asymptotic properties of polynomials -- 1. Finite representability and ultraproducts -- 2. Spreading models -- 3. Polynomials and p-estimates -- 4. Separating polynomials. Symmetric and sub-symmetric polynomials -- 5. Stabilisation of polynomials -- 6. Sub-symmetric polynomials on Rn -- 7. Polynomial algebras on Banach spaces -- 8. Notes and remarks -- Chapter 5. Smoothness and structure -- 1. Convex functions -- 2. Smooth bumps and structure I -- 3. Smooth variational principles -- 4. Smooth bumps and structure II -- 5. Local dependence on finitely many coordinates -- 6. Isomorphically polyhedral spaces -- 7. L_p spaces -- 8. C(K) spaces -- 9. Orlicz spaces -- 10. Notes and remarks -- Chapter 6. Structural behaviour of smooth mappings -- 1. Weak uniform continuity and higher smoothness -- 2. Bidual extensions -- 3. Class ==========W -- 4. Uniformly smooth mappings from C(K), K scattered
  • 5. Uniformly smooth mappings from ==========W-spaces -- 6. Fixing the canonical basis of c_0 -- 7. Ranges of smooth mappings -- 8. Harmonic behaviour of smooth mappings -- 9. Notes and remarks -- Chapter 7. Smooth approximation -- 1. Separation -- 2. Approximation by polynomials -- 3. Approximation by real-analytic mappings -- 4. Infimal convolution -- 5. Approximation of continuous mappings and partitions of unity -- 6. Non-linear embeddings into c_0() -- 7. Approximation of Lipschitz mappings -- 8. Approximation of C1-smooth mappings -- 9. Approximation of norms -- 10. Notes and remarks -- Bibliography -- Notation -- Index
  • Chapter 5. Smoothness and structure
  • Chapter 7. Smooth approximation
  • Chapter 2. Basic properties of polynomials on Rn
  • Index
  • Notation
  • -
  • Chapter 4. Asymptotic properties of polynomials
  • /
  • Chapter 6. Structural behaviour of smooth mappings
  • Contents
  • Introduction
  • Frontmatter --
  • Chapter 1. Fundamental properties of smoothness
  • Chapter 3. Weak continuity of polynomials and estimates of coefficients
  • Bibliography