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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Main Authors: Hájek, Petr, Johanis, Michal
Format: eBook Book
Language:English
Published: Germany De Gruyter 2014
De Gruyter, Inc
Edition:1
Series:De Gruyter Series in Nonlinear Analysis and Applications
Subjects:
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
Online Access:Get full text
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Abstract 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 around two main broad questions: What is the best smoothness of a given Banach space, and its structural consequences? How large is a supply of smooth functions in the sense of approximating continuous functions in the uniform topology, i.e. how does the Stone-Weierstrass theorem generalize into infinite dimension where measure and compactness are not available? The subject of infinite dimensional real higher smoothness is treated here for the first time in full detail, therefore this book may also serve as a reference book.
AbstractList 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 around two main broad questions: What is the best smoothness of a given Banach space, and its structural consequences? How large is a supply of smooth functions in the sense of approximating continuous functions in the uniform topology, i.e. how does the Stone-Weierstrass theorem generalize into infinite dimension where measure and compactness are not available? The subject of infinite dimensional real higher smoothness is treated here for the first time in full detail, therefore this book may also serve as a reference book.
The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome. Please submit book proposals toJürgen Appell.
This bookis aboutthe 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 around two main broad questions: What is the best smoothness of a given Banach space, and its structural consequences? How large is a supply of smooth functions in the sense of approximating continuous functions in the uniform topology, i.e. how does the Stone-Weierstrass theorem generalize into infinite dimension where measure and compactness are not available? The subject of infinite dimensional real higher smoothness is treatedherefor the first time in full detail, therefore this book may also serve as a reference book.
This bookis aboutthe 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 around two main broad questions: What is the best smoothness of a given Banach space, and its structural consequences? How large is a supply of smooth functions in the sense of approximating continuous functions in the uniform topology, i.e. how does the Stone-Weierstrass theorem generalize into infinite dimension where measure and compactness are not available? The subject of infinite dimensional real higher smoothness is treatedherefor the first time in full detail, therefore this book may also serve as a reference book.
Author Hájek, Petr
Johanis, Michal
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ISBN 3110258994
9783110258998
9783110258981
3110258986
9783110391992
3110391996
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Keywords Polynomial
Banach Space
Approximation
Variational Principle
Smoothness
LCCN 2014017465
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Language English
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Notes Includes bibliographical references (p. [467]-489) and index
OCLC 898769617
PQID EBC1130389
PageCount 513 pages
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Snippet 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...
The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various...
This bookis aboutthe subject of higher smoothness in separable real Banach spaces.It brings together several angles of view on polynomials, both in finite and...
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SubjectTerms Approximation
Banach Space
Banach spaces
MATHEMATICS
MATHEMATICS / Functional Analysis
MATHEMATICS / Mathematical Analysis
MATHEMATICS / Reference
Mathematics. Analysis
Normed linear spaces
Polynomial
Polynomials
Smoothness
Variational Principle
TableOfContents 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
Title Smooth Analysis in Banach Spaces
URI http://digital.casalini.it/9783110258998
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Volume 19
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