Table Of ContentJürgen Jost
Postmodern Analysis
ThirdEdition
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JürgenJost
MaxPlanckInstitute
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To Eberhard Zeidler,
for building up our Institute with so
much enthusiasm and human skill
Preface
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 ap-
proach to analysis presented here from what has by its protagonists been
called“ModernAnalysis”.“ModernAnalysis”asrepresentedintheworksof
the Bourbaki group or in the textbooks by Jean Dieudonn´e 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
tobenecessarytoconcentratesolelyontheinternalproblemsandstructures
and to neglect the relations to other fields of scientific, even of mathematical
studyforacertainwhile.Almostcompleteisolationmayberequiredtoreach
the level of intellectual elegance and perfection that only a good mathemat-
ical 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. The axiomatic approach started by Hilbert and taken up
and perfected by the Bourbaki group has led to some of the most important
mathematical contributions of our century, most notably in the area of al-
gebraic geometry. This development was definitely beneficial for many areas
of mathematics, but for other fields this was not true to the same extent. In
geometry, the powerful tool of visual imagination was somewhat neglected,
and global nonlinear phenomena connected with curvature could not always
be addressed adequately. In analysis, likewise, perhaps too much emphasis
was laid on the linear theory, while the genuinely nonlinear problems were
found to be too diverse to be subjected to a systematic and encompassing
theory.Thiseffectwasparticularlynoticableinthefieldofpartialdifferential
equations. This branch of mathematics is one of those that have experienced
the most active and mutually stimulating interaction with the sciences, and
those equations that arise in scientific applications typically exhibit some
genuinely nonlinear structure because of self-interactions and other effects.
Thus, modern mathematics has been concerned with its own internal
structure, and it has achieved great successes there, but perhaps it has lost
VIII Preface
a little of the stimulation that a closer interaction with the sciences can of-
fer. This trend has been reversed somewhat in more recent years, and in
particular rather close ties have been formed again between certain areas of
mathematicsandtheoreticalphysics.Also,inmathematicalresearch,theem-
phasis has perhaps shifted a bit from general theories back to more concrete
problems that require more individual methods.
Ithereforefeltthatitwould beappropriatetopresentan introduction to
advancedanalysisthatpreservesthedefiniteachievementsofthetheorythat
callsitself“modern”,butatthesametimetranscendsthelatter’slimitations.
For centuries, “modern” in the arts and the sciences has always meant
“new”,“differentfromtheancient”,sometimeseven“revolutionary”,andso
itwasanepithetthatwasconstantlyshiftingfromoneschooltoitssuccessor,
and it never stuck with any artistic style or paradigm of research. That only
changedinourcentury,whenabstractfunctionalitywascarriedtoitsextreme
in architecture and other arts. Consequently, in a certain sense, any new
theory or direction could not advance any further in that direction, but had
totakesomestepsbackandtakeupsomeoftheachievementsof“premodern”
theories. Thus, the denomination “modern” became attached to a particular
style and the next generation had to call itself “postmodern”. As argued
above,thesituationinmathematicsisincertainregardscomparabletothat,
anditthusseemslogicaltocall“postmodern”anapproachthattriestobuild
upon the insights of the modern theory, but at the same time wishes to take
back the latter’s exaggerations.
Ofcourse,theword“postmodern”doesnotatpresenthaveanaltogether
positive meaning as it carries some connotations of an arbitrary and unprin-
cipled mixture of styles. Let me assure the potential reader that this is not
intended by the title of the present book. I wish rather to give a coherent
introduction to advanced analysis without abstractions for their own sake
that builds a solid basis for the areas of partial differential equations, the
calculus of variations, functional analysis and other fields of analysis, as well
as for their applications to analytical problems in the sciences, in particular
the ones involving nonlinear effects.
Ofcourse,calculusisbasicforallofanalysis,butmoretothepoint,there
seem to be three key theories that mathematical analysis has developed in
ourcentury,namelytheconceptofBanachspace,theLebesgueintegral,and
the notion of abstract differentiable manifold. Of those three, the first two
are treated in the present book, while the third one, although closest to the
author’s own research interests, has to wait for another book (this is not
quite true, as I did treat that topic in more advanced books, in particular in
“Riemannian Geometry and Geometric Analysis”, Springer, 1995).
The Lebesgue integration theory joins forces with the concept of Banach
spaces when the Lp and Sobolev spaces are introduced, and these spaces are
basic tools for the theory of partial differential equations and the calculus of
variations. (In fact, this is the decisive advantage of the Lebesgue integral
Preface IX
over the older notion, the so-called Riemann integral, that it allows the con-
struction of complete normed function spaces, i.e. Hilbert or Banach spaces,
namely the Lp and Sobolev spaces.) This is the topic that the book will lead
the reader to.
The organization of the book is guided by pedagogical principles. After
all,itoriginatedinacourseItaughttostudentsinBochumatthebeginning
level of a specialized mathematics education. Thus, after carefully collect-
ing the prerequisites about the properties of the real numbers, we start with
continuous functions and calculus for functions of one variable. The intro-
duction of Banach spaces is motivated by questions about the convergence
of sequences of continuous or differentiable functions. We then develop some
notionsaboutmetricspaces,andtheconceptofcompactnessreceivespartic-
ular attention. Also, after the discussion of the one-dimensional theory, we
hope that the reader is sufficiently prepared and motivated to be exposed
to the more general treatment of calculus in Banach spaces. After present-
ingsomeratherabstractresults,thediscussionbecomesmoreconcreteagain
with calculus in Euclidean spaces. The implicit function theorem and the
Picard-Lindelo¨f theorem on the existence and uniqueness of solutions of or-
dinarydifferentialequations(ODEs)areboth derived fromtheBanach fixed
point theorem.
In the second part, we develop the theory of the Lebesgue integral in Eu-
clidean spaces. As already mentioned, we then introduce Lp and Sobolev
spaces and give an introduction to elliptic partial differential equations
(PDEs) and the calculus of variations. Along the way, we shall see several
examples arising from physics.
In the Table of Contents, I have described the key notions and results of
each section, and so the interested reader can find more detailed information
about the contents of the book there.
Thisbookpresentsanintermediateanalysiscourse.Thus,itslevelissome-
what higher than the typical introductory courses in the German university
system. Nevertheless, in particular in the beginning, the choice and presen-
tation of material are influenced by the requirement of such courses, and
I have utilized some corresponding German textbooks, namely the analysis
courses of 0. Forster (Analysis I - III, Vieweg 1976ff.) and H. Heuser (Anal-
ysis I, II, Teubner, 1980ff.). Although the style and contents of the present
book are dictated much more by pedagogical principles than is the case in
J. Dieudonn´e’s treatise Modern Analysis, Academic Press, 1960ff., there is
some overlap of content. Although typically the perspective of my book is
different, the undeniable elegance of reasoning that can be found in the trea-
tise of Dieudonn´e nevertheless induced me sometimes to adapt some of his
arguments, in line with my general principle of preserving the achievements
of the theory that called itself modern.
ForthetreatmentofSobolevspacesandtheregularityofsolutionsofellip-
ticpartialdifferentialequations,IhaveusedD.Gilbarg,N.Trudinger,Elliptic
X Preface
Partial Differential Equations of Second Order, Springer, 21983, although of
course the presentation here is more elementary than in that monograph.
I received the competent and dedicated help – checking the manuscript
for this book, suggesting corrections and improvements, and proofreading –
of Felicia Bernatzki, Christian Gawron, Lutz Habermann, Xiaowei Peng,
MonikaReimpell,WilderichTuschmann,andTilmannWurzbacher.Myorig-
inal German text was translated into English by Hassan Azad. The typing
and retyping of several versions of my manuscript was performed with pa-
tience and skill by Isolde Gottschlich. The figures were created by Micaela
Krieger with the aid of Harald Wenk and Ralf Muno. I thank them all for
their help without which this book would not have become possible.
Preface to the 2nd edition
For this edition, I have added some material on the qualitative behavior of
solutions of ordinary differential equations, some further details on Lp and
Sobolev functions, partitions of unity, and a brief introduction to abstract
measure theory. I have also used this opportunity to correct some misprints
and two errors from the first edition. I am grateful to Horst Lange, C. G.
Simader, and Matthias Stark for pertinent comments. I also should like to
thank Antje Vandenberg for her excellent TEXwork.
Preface to the 3rd edition
This edition corrects some misprints and minor inconsistencies that were
kindly pointed out to me by several readers, in particular Bruce Gould, as
wellasanerrorintheproofoftheorem19.16thatwasbroughttomyattention
by Matthias Stark. I have also used this opportunity to introduce another
important tool in analysis, namely covering theorems. Useful references for
such results and further properties of various classes of weakly differentiable
functions are W.Ziemer, Weakly differentiable functions, Springer, 1989, and
L.C.Evans, R.Gariepy, Measure theory and fine properties of functions, CRC
Press,1992,aswellasthefundamentalH.Federer,Geometricmeasuretheory,
Springer, 1969.
Leipzig, January 2005 Ju¨rgen Jost
Contents
Chapter I. Calculus for Functions of One Variable
0. Prerequisites
Propertiesoftherealnumbers,limitsandconvergenceofsequencesof
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(cid:1) = 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
Definitionofametricspace,open,closed,convex,connected,compact
sets,sequentialcompactness,continuousmappingsbetweenmetricspaces,
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
