# Postmodern Analysis

###### Jürgen Jost
Publisher:
Springer Verlag
Publication Date:
2005
Number of Pages:
371
Format:
Paperback
Edition:
3
Series:
Universitext
Price:
49.95
ISBN:
3-540-25830-2
Category:
Textbook
[Reviewed by
Fernando Q. Gouvêa
, on
01/20/2006
]

The title of Jost's Postmodern Analysis will probably entice some readers while scaring others away. It certainly enticed me. So let's start by considering what Jost is trying to do.

The modern period in the arts was characterized by high levels of abstraction and a lack of interest in the functions of art in society. To call a piece "decorative" was an insult. Similarly, Jost describes modern analysis as highly abstract, focused on its own internal structure and development, and not really attentive to applications. He argues in the introduction that such an emphasis was needed and useful when it was first introduced, since analysis texts seemed to have degenerated into collections of unrelated tricks. On the other hand, it had its problems, the most obvious of which was the focus on linear problems (which were amenable to the high-level theory then in vogue) and a lack of attention to non-linear problems (which were not).

We are now, says Jost, in the postmodern period. By this he means that analysis has had to backtrack a little from its focus on abstraction and structure and pay more attention to the (interesting but very difficult) problems coming from applied mathematics. He emphasizes that he does not mean that his book displays "an arbitrary and unprincipled mixture of styles" (which is one of the meanings of "postmodern" in the arts). Neither, of course, does it embody the postmodern disbelief in truth (it'd be hard to do mathematics that way). But it does, in a way, share the postmodern disinterest in "metanarratives" — after all, if Bourbaki's Elements or Dieudonné's Modern Analysis aren't metanarrative, what is?

Well, does Jost pull it off? As the table of contents shows, this is a (graduate level) intermediate analysis text. Jost considers the theory of Banach spaces and the Lebesgue integral to be two of the three "key theories" developed in the 20th century (the third is the theory of differentiable manifolds), so the book emphasizes those themes and their connection to the Calculus of Variations and Partial Differential Equations. I find this a reasonable choice of pathway through the subject. It seems to me, however, that this is more a change in emphasis than a serious break with the past.

Stylistically, the book strikes me as no less abstract than most "modern" books were. (That's true about much postmodern art too!) It is written in a mostly uncompromising definition-theorem-proof style. Most graduate students, I think, would find it hard going, and students interested in applied mathematics should look elsewhere unless they are capable of quite a bit of delayed gratification, since while there is a lot of applicable mathematics here, there are very few actual applications.

Fernando Q. Gouvêa is professor of mathematics at Colby College.

Chapter I. Calculus for Functions of One Variable

0. Prerequisites

Properties of the real numbers, limits and convergence of sequences of

real numbers, exponential function and logarithm. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1. Limits and Continuity of Functions

Definitions of continuity, uniform continuity, properties of continuous

functions, intermediate value theorem, H¨older and Lipschitz continuity.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2. Differentiability

Definitions of differentiability, differentiation rules, differentiable

functions are continuous, higher order derivatives. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3. Characteristic Properties of Differentiable Functions.

Differential Equations

Characterization of local extrema by the vanishing of the derivative,

mean value theorems, the differential equation f = γf, uniqueness of

solutions of differential equations, qualitative behavior of solutions of

differential equations and inequalities, characterization of local maxima

and minima via second derivatives, Taylor expansion. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4. The Banach Fixed Point Theorem. The Concept of

Banach Space

Banach fixed point theorem, definition of norm, metric, Cauchy

sequence, completeness. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

XII Contents

5. Uniform Convergence. Interchangeability of Limiting

Processes. Examples of Banach Spaces. The Theorem

of Arzela-Ascoli

Convergence of sequences of functions, power series, convergence

theorems, uniformly convergent sequences, norms on function spaces,

theorem of Arzela-Ascoli on the uniform convergence of sequences of

uniformly bounded and equicontinuous functions. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6. Integrals and Ordinary Differential Equations

Primitives, Riemann integral, integration rules, integration by parts,

chain rule, mean value theorem, integral and area, ODEs, theorem of

Picard-Lindel¨of on the local existence and uniqueness of solutions of

ODEs with a Lipschitz condition. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Chapter II. Topological Concepts

7. Metric Spaces: Continuity, Topological Notions,

Compact Sets

Definition of a metric space, open, closed, convex, connected, compact

sets, sequential compactness, continuous mappings between metric spaces,

bounded linear operators, equivalence of norms in Rd, definition of a

topological space. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Chapter III. Calculus in Euclidean and Banach Spaces

8. Differentiation in Banach Spaces

Definition of differentiability of mappings between Banach spaces,

differentiation rules, higher derivatives, Taylor expansion. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

9. Differential Calculus in Rd

A. Scalar valued functions

Gradient, partial derivatives, Hessian, local extrema, Laplace

operator, partial differential equations

B. Vector valued functions

Jacobi matrix, vector fields, divergence, rotation. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

10. The Implicit Function Theorem. Applications

Implicit and inverse function theorems, extrema with constraints,

Lagrange multipliers. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Contents XIII

11. Curves in Rd. Systems of ODEs

Regular and singular curves, length, rectifiability, arcs, Jordan arc

theorem, higher order ODE as systems of ODEs. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Chapter IV. The Lebesgue Integral

12. Preparations. Semicontinuous Functions

Theorem of Dini, upper and lower semicontinuous functions, the

characteristic function of a set. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

13. The Lebesgue Integral for Semicontinuous Functions.

The Volume of Compact Sets

The integral of continuous and semicontinuous functions, theorem of

Fubini, volume, integrals of rotationally symmetric functions and other

examples. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

14. Lebesgue Integrable Functions and Sets

Upper and lower integral, Lebesgue integral, approximation of

Lebesgue integrals, integrability of sets. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

15. Null Functions and Null Sets. The Theorem of Fubini

Null functions, null sets, Cantor set, equivalence classes of integrable

functions, the space L1, Fubini’s theorem for integrable functions.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

16. The Convergence Theorems of Lebesgue Integration

Theory

Monotone convergence theorem of B. Levi, Fatou’s lemma, dominated

convergence theorem of H. Lebesgue, parameter dependent integrals,

differentiation under the integral sign. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

17. Measurable Functions and Sets. Jensen’s Inequality.

The Theorem of Egorov

Measurable functions and their properties, measurable sets,

measurable functions as limits of simple functions, the composition of a

measurable function with a continuous function is measurable, Jensen’s

inequality for convex functions, theorem of Egorov on almost uniform

convergence of measurable functions, the abstract concept of a measure.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

XIV Contents

18. The Transformation Formula

Transformation of multiple integrals under diffeomorphisms, integrals

in polar coordinates. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Chapter V. Lp and Sobolev Spaces

19. The Lp-Spaces

Lp-functions, H¨older’s inequality, Minkowski’s inequality,

completeness of Lp-spaces, convolutions with local kernels, Lebesgue

points, approximation of Lp-functions by smooth functions through

mollification, test functions, covering theorems, partitions of unity.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

20. Integration by Parts. Weak Derivatives. Sobolev Spaces

Weak derivatives defined by an integration by parts formula, Sobolev

functions have weak derivatives in Lp-spaces, calculus for Sobolev

functions, Sobolev embedding theorem on the continuity of Sobolev

functions whose weak derivatives are integrable to a sufficiently high

power, Poincar´e inequality, compactness theorem of Rellich-Kondrachov

on the Lp-convergence of sequences with bounded Sobolev norm.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Chapter VI. Introduction to the Calculus of Variations and

Elliptic Partial Differential Equations

21. Hilbert Spaces. Weak Convergence

Definition and properties of Hilbert spaces, Riesz representation

theorem, weak convergence, weak compactness of bounded sequences,

Banach-Saks lemma on the convergence of convex combinations of

bounded sequences. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

22. Variational Principles and Partial Differential Equations

Dirichlet’s principle, weakly harmonic functions, Dirichlet

problem, Euler-Lagrange equations, variational problems, weak lower

semicontinuity of variational integrals with convex integrands, examples

from physics and continuum mechanics, Hamilton’s principle, equilibrium

states, stability, the Laplace operator in polar coordinates. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

23. Regularity of Weak Solutions

Smoothness of weakly harmonic functions and of weak solutions of

general elliptic PDEs, boundary regularity, classical solutions. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Contents XV

24. The Maximum Principle

Weak and strong maximum principle for solutions of elliptic PDEs,

boundary point lemma of E. Hopf, gradient estimates, theorem of

Liouville. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

25. The Eigenvalue Problem for the Laplace Operator

Eigenfunctions of the Laplace operator form a complete orthonormal

basis of L2 as an application of the Rellich compactness theorem.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365