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1998 | Book

Prerequisites Read chapter Read first chapter

Postmodern Analysis

Author: Jürgen Jost

Publisher: Springer Berlin Heidelberg

Book Series : Universitext

Included in: Professional Book Archive

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About this book

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 approach to analysis presented here from what has been called "Modern Analysis" by its protagonists. "Modern Analysis" as represented in the works of the Bour­ baki 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 degen­ erate into a collection of rather unconnected tricks to solve special problems, this definitively represented a healthy achievement. In any case, for the de­ velopment of a consistent and powerful mathematical theory, it seems to be necessary to concentrate solelyon the internal problems and structures and to neglect the relations to other fields 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 might be useful to open one's eyes again to the inspiration coming from concrete ex­ ternal problems.

Table of Contents

Frontmatter

Calculus for Functions of One Variable

Frontmatter
0. Prerequisites
Abstract
We review some basic material, in particular the convergence of sequences of real numbers, and also properties of the exponential function and the logarithm.
Jürgen Jost
1. Limits and Continuity of Functions
Abstract
We introduce the concept of continuity for a function defined on a subset of ℝ (or ℂ). After deriving certain elementary properties of continuous functions, we show the intermediate value theorem, and that a continuous function defined on a closed and bounded set assumes its maximum and minimum there.
Jürgen Jost
2. Differentiability
Abstract
We define the notion of differentiability of functions defined on subsets of ℝ, and we show the basic rules for the computation of derivatives.
Jürgen Jost
3. Characteristic Properties of Differentiable Functions. Differential Equations
Abstract
We treat the mean value theorems for differentiable functions, characterize interior minima and maxima of such functions in terms of properties of the derivatives, discuss some elementary aspects of differential equations, and finally show Taylor’s formula.
Jürgen Jost
4. The Banach Fixed Point Theorem. The Concept of Banach Space
Abstract
As the proper setting for the convergence theorems of subsequent §§, we introduce the concept of a Banach space as a complete normed vector space. The Banach fixed point theorem is discussed in detail.
Jürgen Jost
5. Uniform Convergence. Interchangeability of Limiting Processes. Examples of Banach Spaces. The Theorem of Arzela-Ascoli
Abstract
We introduce the notion of uniform convergence. This leads to Banach spaces of continuous and differentiable functions. We discuss when the limit of the derivatives of a convergent sequence of functions equals the derivative of the limit and related questions. The theorem of Arzela-Ascoli is shown, saying that an equicontinuous and uniformly bounded sequence of functions on a closed and bounded set contains a uniformly convergent subsequence.
Jürgen Jost
6. Integrals and Ordinary Differential Equations
Abstract
A continuous function g is called a primitive of another function f if the derivative of g exists and coincides with f. A primitive thus is an indefinite integral. We derive the basic rules for the computation of integrals. We use the Banach fixed point theorem to derive the Picard-Lindelöf theorem on the local existence of solutions or integrals of ordinary differential equations (ODEs).
Jürgen Jost

Topological Concepts

Frontmatter
7. Metric Spaces: Continuity, Topological Notions, Compact Sets
Abstract
We introduce the elementary topological concepts for metric spaces, like open, closed, and compact subsets. Continuity of functions is also expressed in topological terms. At the end, we also briefly discuss abstract topological spaces.
Jürgen Jost

Calculus in Euclidean and Banach Spaces

Frontmatter
8. Differentiation in Banach Spaces
Abstract
We introduce the concept of differentiability for mappings between Banach spaces, and we derive the elementary rules for differentiation.
Jürgen Jost
9. Differential Calculus in ℝ d
Abstract
The results of the previous paragraph are specialized to Euclidean spaces. Again, as in §3, interior extrema of differentiable functions are studied. We also introduce some standard differential operators like the Laplace operator.
Jürgen Jost
10. The Implicit Function Theorem. Applications
Abstract
The Banach fixed point theorem is used to derive the implicit function theorem. Corollaries are the inverse function theorem and the Lagrange multiplier rules for extrema with side conditions.
Jürgen Jost
11. Curves in ℝd. Systems of ODEs
Abstract
First, some elementary properties, like rectifiability or arc length parametrization, of curves in Euclidean space are treated. Next, curves that solve systems of ODEs are considered. Higher order ODEs are reduced to such systems.
Jürgen Jost

The Lebesgue Integral

Frontmatter
12. Preparations. Semicontinuous Functions
Abstract
As a preparation for Lebesgue integration theory, lower and upper semicontinuous functions are studied.
Jürgen Jost
13. The Lebesgue Integral for Semicontinuous Functions. The Volume of Compact Sets
Abstract
We define the integral of semicontinuous functions, and consider properties of such integrals, like Fubini’s theorem. Volumes of compact sets are defined, and certain rules, like Cavalieri’s principle, for their computation are given. In particular, computations simplify in rotationally symmetric situations.
Jürgen Jost
14. Lebesgue Integrable Functions and Sets
Abstract
The general Lebesgue integral is defined, and basic properties are derived. Here, a function is called Lebesgue integrable if approximation from above by lower semicontinuous functions leads to the same result as approximation from below by upper semicontinuous functions. Sets are called integrable when their characteristic functions are.
Jürgen Jost
15. Null Functions and Null Sets. The Theorem of Fubini
Abstract
Null functions, i.e. those whose integral is 0 on every set, and null sets, i.e. those whose volume vanishes, are negligible for purposes of integration theory. In particular, countable sets, or lower dimensional subsets of Euclidean spaces are null sets. The general theorem of Fubini saying that in multiple integrals the order of integration is irrelevant is shown.
Jürgen Jost
16. The Convergence Theorems of Lebesgue Integration Theory
Abstract
We discuss the fundamental convergence theorems of Fatou, B. Levi, and Lebesgue, saying that under certain assumptions, the integral of a limit of a sequence of functions equals the limit of the integrals. Instructive examples show the necessity of those assumptions. As an application, results that justify the derivation under the integral sign w.r.t. a parameter are given.
Jürgen Jost
17. Measurable Functions and Sets. Jensen’s Inequality. The Theorem of Egorov
Abstract
We introduce the general notion of a measurable function and a measurable set. Measurable functions are characterized as pointwise limits of finite valued functions. Jensen’s inequality for the integration of convex functions and Egorov’s theorem saying that an almost everywhere converging sequence of functions also converges almost uniformly, i.e. uniformly except on a set of arbitrarily small measure, are derived.
Jürgen Jost
18. The Transformation Formula
Abstract
The general transformation formula for multiple integrals is derived. Transformation from Euclidean to polar coordinates is discussed in detail.
Jürgen Jost

L p and Sobolev Spaces

19. The L p -Spaces
Abstract
The L p -spaces of functions the pth power of whose absolute value is integrable are introduced. Minkowski’s and Hölder’s inequality are shown, and the L p -spaces are seen to be Banach spaces. Also, the approximation of L p -functions by smooth ones is discussed.
Jürgen Jost
20. Integration by Parts. Weak Derivatives. Sobolev Spaces
Abstract
Weak derivatives are introduced by taking the rule for integration by parts as a definition. Spaces of functions that are in L p together with certain weak derivatives are called Sobolev spaces. Sobolev’s embedding theorem says that such functions are continuous if their weak derivatives satisfy strong enough integrability properties. Rellich’s compactness theorem says that integral bounds on weak derivatives imply convergence of subsequences of the functions itself in L p .
Jürgen Jost

Introduction to the Calculus of Variations and Elliptic Partial Differential Equations

Frontmatter
21. Hilbert Spaces. Weak Convergence
Abstract
Hilbert spaces are Banach spaces with norm derived from a scalar product. A sequence in a Hilbert space is said to converge weakly if its scalar product with any fixed element of the Hilbert space converges. Weak convergence satisfies important compactness properties that do not hold for ordinary convergence in an infinite dimensional Hilbert space. In particular, any bounded sequence contains a weakly convergent subsequence.
Jürgen Jost
22. Variational Principles and Partial Differential Equations
Abstract
Dirichlet’s principle consists in constructing harmonic functions by minimizing the Dirichlet integral in an appropriate class of functions. This idea is generalized, and minimizers of variational integrals are weak solutions of the associated differential equations of Euler and Lagrange. Several examples are discussed.
Jürgen Jost
23. Regularity of Weak Solutions
Abstract
It is shown that under appropriate ellipticity assumptions, weak solutions of partial differential equations (PDEs) are smooth. This applies in particular to the Laplace equation for harmonic functions, thereby justifying Dirichlet’s principle introduced in the previous paragraph.
Jürgen Jost
24. The Maximum Principle
Abstract
The strong maximum principle of E. Hopf says that a solution of an elliptic PDE cannot assume an interior maximum. This leads to further results about solutions of such PDEs, like removability of singularities, gradient bounds, or Liouville’s theorem saying that every bounded harmonic functions defined on all of Euclidean space is constant.
Jürgen Jost
25. The Eigenvalue Problem for the Laplace Operator
Abstract
We use Rellich’s embedding theorem to show that every L 2 function on an open ,fl Ω ⊂ ℝ d can be expanded in terms of eigenfunctions of the Laplace operator on Ω.
Jürgen Jost
Backmatter
Metadata
Title
Postmodern Analysis
Author
Jürgen Jost
Copyright Year
1998
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-03635-8
Print ISBN
978-3-540-63485-0
DOI
https://doi.org/10.1007/978-3-662-03635-8