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Postmodern Analysis
Author
Title
Postmodern Analysis [electronic resource] / by Jürgen Jost.
ISBN
9783540288909
Edition
Third Edition.
Publication
Berlin, Heidelberg : Springer Berlin Heidelberg, 2005.
Physical Description
1 online resource (XV, 375 p).
Local Notes
Access is available to the Yale community.
Access and use
Access restricted by licensing agreement.
Summary
What is the title of this book intended to signify, what connotations is the adjective “Postmodern” meant to carry? A potential reader will surely pose this question. To answer it, I should describe what distinguishes the appproach to analysis presented here from what has by its protagonists been called “Modern Analysis”. “Modern Analysis” as represented in the works of the Bourbaki group or in the textbooks by Jean Dieudonne is characterized by its systematic and axiomatic treatment and by its drive towards a high level of abstraction. Given the tendency of many prior treatises on analysis to degenerate into a collection of rather unconnected tricks to solve special problems, this definitely represented a healthy achievement. In any case, for the development of a consistent and powerful mathematical theory, it seems to be necessary to concentrate solely on the internal problems and structures and to neglect the relations to other felds of scientific, even of mathematical study for a certain while. Almost complete isolation may be required to reach the level of intellectual elegance and perfection that only a good mathematical theory can acquire. However, once this level has been reached, it can be useful to open one’s eyes again to the inspiration coming from concrete external problems.
Variant and related titles
Springer ENIN.
Other formats
Printed edition:
Printed edition:
Format
Books / Online
Language
English
Added to Catalog
April 03, 2019
Series
Universitext,
Contents
Calculus for Functions of One Variable
Prerequisites
Limits and Continuity of Functions
Differentiability
Characteristic Properties of Differentiable Functions. Differential Equations
The Banach Fixed Point Theorem. The Concept of Banach Space
Uniform Convergence. Interchangeability of Limiting Processes. Examples of Banach Spaces. The Theorem of Arzela-Ascoli
Integrals and Ordinary Differential Equations
Topological Concepts
Metric Spaces: Continuity, Topological Notions, Compact Sets
Calculus in Euclidean and Banach Spaces
Differentiation in Banach Spaces
Differential Calculus in $$\mathbb{R}$$ d
The Implicit Function Theorem. Applications
Curves in $$\mathbb{R}$$ d. Systems of ODEs
The Lebesgue Integral
Preparations. Semicontinuous Functions
The Lebesgue Integral for Semicontinuous Functions. The Volume of Compact Sets
Lebesgue Integrable Functions and Sets
Null Functions and Null Sets. The Theorem of Fubini
The Convergence Theorems of Lebesgue Integration Theory
Measurable Functions and Sets. Jensen’s Inequality. The Theorem of Egorov
The Transformation Formula
and Sobolev Spaces
The Lp-Spaces
Integration by Parts. Weak Derivatives. Sobolev Spaces
to the Calculus of Variations and Elliptic Partial Differential Equations
Hilbert Spaces. Weak Convergence
Variational Principles and Partial Differential Equations
Regularity of Weak Solutions
The Maximum Principle
The Eigenvalue Problem for the Laplace Operator.
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