Real Analysis with Economic Applications - Content Preface pps

.
REAL ANALYSIS
with
ECONOMIC APPLICAT IONS
EFE A. OK
New York Universit y
December, 2005
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mat he matics is very much like poetry what m akes a good poem - a
great poem - i s that there is a large amount of thought expressed in very
few word s . I n t his sense formulas like
e
πi
+1=0 or
]
∞
−∞
e
−x
2
dx =
√
π
are poems.
Lipman Bers
ii
Con ten ts
Preface
Cha p ter A Prelim inaries o f Real Analys is
A.1 El emen ts of Set Theory
1 Sets

2 Relations
3 Equivalence Relations
4 Order Relations
5 Functions
6 Sequences, Vectors and M atrices
7
∗
A Glimpse of Advanced Set Theory: The Axiom of Choice
A.2 Real Numbers
1 Ordered Fields
2 Natural Numbers, Integers and Rationals
3 Real Numbers
4 Intervals and
R
A.3 Real Sequences
1 Convergent Sequences
2 Monotonic Sequences
3 Subsequential Limits
4 Infinite Series
5 Rearrangements of Infinite Series
6 Infinite Products
A.4 Real Functions
1 Basic Definitions
2 Limits,ContinuityandDifferentiation
3 Riemann Integration
4 Exponential, Logarithmic and Trigonometric Functions
5 Concave and Convex Functions
6 Quasiconcave and Quasiconvex Functions
Chapter B Coun tability
B.1 Countable and Uncoun table Sets

B.2 Los ets and Q
B.3 Some More Advanced Theory
1 The Cardinality Ordering
2
∗
The Well Ordering Principle
iii
B.4 Application: O r d in al U tility T he ory
1 Preference Relations
2 Utility Representation of Complete Preference Relations
3
∗
Utility Representation o f Incomplete Preference Relations
Chapter C Metric Spaces
C.1 Ba sic Notions
1 Metric Spaces: Definitions and Examples
2 Open and Closed Sets
3 Convergent Sequences
4 Sequential Characterization of Closed Sets
5 Equivalence of Metrics
C.2 Connec ted n e ss and Separability
1 Connected Metric Spaces
2 Separable Metric Spaces
3 Applications to Utility Theory
C.3 Co mpactness
1 Basic Definitions and the Heine-Borel Theorem
2 CompactnessasaFiniteStructure
3 Closed and Bounded Sets
C.4 Sequen tial Compactness
C.5 Co mpleteness

1 Cauchy Sequences
2 Complete Metric Spaces: Definition and Examples
3 Completeness vs. Closedness
4 Completeness vs. Compactness
C.6 Fixe d Point Theory I
1 Contractions
2 The Banach Fix ed Point Theorem
3
∗
Generalizations of the Banac h Fixed Point Theorem
C.7 Applications to Function al Equations
1 Solutions of Fu nctional Equations
2 Picard’s Existence Theorems
C.8 Products of Metric Spaces
1 Finite Products
2 Countably Infinite Products
Chapter D Continuit y I
D.1 Co ntinuity of Functions
iv
1 Definitions and Examples
2 Uniform C ontin uity
3 Other Contin uity Concepts
4
∗
Remarks on the Differentiability of Real Functions
5 A Fundamental Characterization of Con tinuity
6 Homeomorphisms
D.2 Continuity and Connectedness
D.3 Continuit y and Compactness
1 Continuous Image of a Compact Set

2 The Local-to-Global Method
3 Weierstrass’ Theorem
D.4 Se micontinuity
D.5 Applications
1
∗
Caristi’s Fixed Poin t Theorem
2 Continuous Representation of a Preference Relation
3
∗
Cauchy’s Functional Equations: Additivity on R
n
4
∗
Representation of Additiv e Preferences
D.6 CB(T ) and Uniform Convergenc e
1 The Basic Metric Structure of CB(T )
2 Uniform Convergence
3
∗
The Stone-Weierstrass Theorem and Separability of C(T )
4
∗
The Arzelà-Ascoli Theorem
D.7
∗
Extension o f Continuous Functions
D.8 Fixe d Point Theory II
1 The Fixed Point Property
2 Retracts

3 The Brou wer Fixed Point Theorem
4 Applications
Chapter E Con tinuit y II
E.1 Correspondences
E.2 Continuity of Correspondences
1 Upper Hemicon tinuity
2 The Closed Graph Property
3 Lo wer Hemicontinuity
4 Continuous Correspondences
5
∗
The Hausdorff Metric and Continuity
E.3 The Maximum Theore m
v
E.4 Application: Stationary Dynamic Programming
1 The Standard Dynamic Programming Problem
2 The P rinciple of Optimality
3 Existence and Uniqueness o f an Optimal Solution
4 Economic Application: The Optimal Growth Model
E.5 Fixed Point Theory III
1 Kakutani’s Fixed Point Theorem
2
∗
Mic hael’s Selection Theorem
3
∗
Proof of Kakutani’s Fixed Point Theorem
4
∗
Contractive Correspondences

E.6 App lication: The Nash Equilibriu m
1 Strategic Games
2 The Nash Equilibrium
3
∗
Remarks on the Equilibria of Discontin uous Games
Chapter F Linear Spaces
F.1 Linear Spaces
1 Abelian Groups
2 Linear Spaces: Definition and Examples
3 Linear Subspaces, Affine Manifolds and Hyperplanes
4 Span and Affine Hull of a S et
5 Linear a nd Affine Independence
6 Bases and Dimension
F.2 L inear Operators and Functionals
1 Definitions and Examples
2 Linear a nd Affine Functions
3 Linear Isomorphisms
4 Hyperplanes, Revisited
F.3 A p p lication: Expected Ut ility Theory
1 The E xpected Utility Theorem
2 Utility Theory under Uncertainty
F.4
∗
Application: Capacities and the Sh apley Va lue
1 Capacities and Coalitional Games
2 The Linear Space of Capacities
3 The Shapley Value
Chapter G Convexit y
G.1 Co nvex Sets

1 Basic Definitions and Examples
2 Convex Cones
vi
3 Ordered Linear Spaces
4 Algebraic and Relative In terior of a Set
5 Algebraic Closure of a Set
6 Finitely Generated Cones
G.2 S epara tion an d E x tension in Linear Spa ces
1 Extension of Linear Functionals
2 Extension of Positive Linear Functionals
3 Separation of Convex Sets by Hyperplanes
4 The External Characterization of Algebraically Closed and Convex Sets
5 Supporting Hyperplanes
6
∗
Superlinear Maps
G.3 Reflection s on R
n
1 Separation in R
n
2 Support in R
n
3 The C auchy-Schwarz Inequalit y
4 Best Ap proximation from a Convex set in
R
n
5 Orthogonal Projections
6 Extension of Positiv e Linear Functionals, Revisited
Cha p te r H Econo m ic App licat ions
H.1 Applica tion s to Expected Utility Th eory

1 The E xpected Multi-Utility Theorem
2
∗
Knigh tian Uncertainty
3
∗
The Gilboa-Schmeidler Multi-Prior Model
H.2 Applications to Welfare Ec onomics
1 The Second Fundamen tal Theorem of Welfare Economics
2 Characterization of Pareto Optima
3
∗
Harsanyi’s Utilitarianism Theorem
H.3 An Application to Information Theory
H.4
∗
Applications to Financial Economics
1 Viability an d Arbitrage-Free P rice Functionals
2 The No-Arbitrage Theorem
H.5 Applications to Cooperative Games
1 The Nash Bargaining Solution
2
∗
Coalitional G ames Without S ide Payments
Chapter I Metric Linear Spaces
I.1 Metric Linear Spaces
I.2 Cont inuous Linear Operators and Functionals
vii
1 Examples of (Dis-)Continuous Linear Operators
2 Contin uity of Positive Linear Functionals

3 Closed vs . Dense Hyperplanes
4 Digression: On the Contin uit y of Concav e Functions
I.3 Finite Dimensional Metric Linear Spaces
I.4
∗
Comp act Sets in Metric Linear Spaces
I.5 Convex Analy sis in Metric Linea r Spaces
1 ClosureandInteriorofaConvexSet
2 Interior vs. Algebraic In terior of a Convex Set
3 Extension of Positiv e Linear Functionals, Revisited
4 Separation by Closed Hyperplanes
5 In terior vs. Algebraic Interior of a Closed and Convex Set
Chapter J Normed Linear Spaces
J.1 Normed Linear Spa ces
1 A G eometric Motivation
2 Normed Linear Spaces
3 Examples of Normed Linear Spaces
4 Metric vs. Normed Linear Spaces
5 Digression: The Lipschitz Continuity of Concave Maps
J.2 Banach Spaces
1 Definition and Examples
2 Infinite Series in Banac h Spaces
3
∗
On the “Size” of Banach Spaces
J.3 Fixed Point T heory IV
1 The G licksberg-Fan Fixed Point T heorem
2 Application: Existence of Nash Equilibrium, Revisited
3
∗

The Schauder Fixed Point Theorems
4
∗
Some Consequences of Schauder’s Theorems
5
∗
Applications to Functional Equations
J.4 Bounded Linear Operators and Functionals
1 Definitions and Examples
2 Linear Homeomorphisms, Revisited
3 The Operator Norm
4 Dual Spaces
5
∗
Discontinuous Linear Functionals, Revisited
J.5 Con vex Analysis in No r med Linear Spaces
1 Separation by Closed Hyperplanes, Revisited
2
∗
Best App roximation from a Convex Set
viii
3 Extreme points
J.6 Extension in Normed Linear Spaces
1 Extension of Continu ous Linear Functionals
2
∗
Infinite Dimensional Normed Linear Spaces
J.7
∗
The Uniform Boundedness Principle

Chapter K Differen tial Calculus
K.1 Fréchet Differen tiation
1 Limits of Functions and Tangency
2 What is a Derivative?
3 The Fréchet Derivative
4 Examples
5 Rules of Differen tiation
6 The Second Fréchet Derivativ e of a Real Function
K.2 Generalizations of the M ean Value Th eorem
1 The Generalized M ean Value Theorem
2
∗
The Mean Value Inequality
K.3 Fréchet Differentiat ion and Concave M aps
1 Remarks -on D ifferen tiability of Concave M aps
2 Fréchet Differentiable Concave Maps
K.4 Optimiza tion
1 Local Extrema of R eal Maps
2 Optimization of Conca ve Maps
K.5 Calculus of Variations
1 Finite Horizon Variational Problems
2 The Euler-Lagrange E quation
3 More on the Sufficiency of the Euler-Lagrange Equation
4 Infinite Horizon Variational Problems
5 Application: The Optimal Investment Problem
6 Application: The Optimal Growth Problem
7 Application: The Poincaré-Wirtinger Inequality
Hin ts For Selected Exercises
References
Index of Symbols

Index of Topics
ix
Preface
This is primarily a textbook on mathem a tical analysis for graduate students in eco-
nomics. While ther e are a large number of excellent textbooks on t his b road topic
in t he mathema tics literature, m ost of these texts are overly a dvanced relative t o the
needs of a vast majority of economics studen ts, and c o ncentrate on various t opics
that a re not readily helpful for studying economic theory. Moreover, it seems that
most economics students lack the time and/or co urage to enroll in a mat h course at
the graduate lev el. Sometim es this is not even for bad rea sons, for only few math
depa rtm ents offer c lasses that are designed fo r the particular needs of economists. Un-
fortunately, m ore o ften th an no t, the consequent lack of mathematical background
creates problems for the students at a later stage of their education since an ex-
ceedingly l arge fraction of economic theory is impenetrable without some rigorous
background in real analysis. The present text aims at providing a rem e dy for this
inconvenient situation.
My treatment is r igorous, yet selective. I prove a good number of results here,
so the reader w ill ha ve plen ty of opportunity to sharpen his/her understanding of
the “theorem-proof” duality, and t o work through a variety of “deep” theorems of
mathem atical analysis. H owever, I take many sh ortcut s. Fo r instance, I av oid com-
plex numbers at a ll c ost, assume compactness of things w hen one could get aw ay w ith
separab ility, introduce topological and/or topological linear con ce p ts only via metrics
and/or norms, and so on . M y objective i s not to report even the m ain theorems i n
their most general form, but rather to giv e a g ood idea to the studen t w h y t hese are
true, or even more im portan tly, w hy one should s uspect tha t they must be true even
before t hey are proved. But the shortcuts a re not ov erly extensive in t he sense t hat
the main results covered here posses s a g ood degree of app licability, especially for
ma in stream economics. Indeed, the purely math ematical developm ent of the text is
put to good use th rou gh sever al a pp lic ation s that provide concise introduction s to a
variety o f topics from economic theory. Among these topics are individual decision

theory, cooperativ e and n oncooperativ e game theory, welfare econ om ics, info rma tion
theory, g eneral equilibrium and finance, and in tertemporal e conomics.
An obv ious dimension that differentiates this text from various books on real
analysis pertains to the choice of topics. I put much more emphasis on topics that
are immediately relevant for economic theory, and omit some standard t hem es of real
analysis that are o f second ary importance for economists. In partic ular , unlike most
treatments of mathematical analysis found in the literature, I work here quite a bit
on order theory, conv ex analy sis, optimization, linear and n o nlin e a r corres pondenc es ,
dynamic programming, and calculus of variations. M oreo ver, apart from direct appli-
cations to economic theory, the exposition includes quite a few fixed point theorems,
along with a leisurely introduction to differential calculus in Banach spaces. (Indeed,
the latter half of the text can be thought of as providing a modest in troduction to
x
geome tric ( n on )lin e ar analysis.) Ho wever, because t h ey play only a m in or role in
modern economic theory, I do not at all discu ss topics like Fourier analysis, Hilbert
spaces and spectral theory in this book.
While I assume here that the student is fa m iliar with the notion of “proof” — within
the first semester of a graduate economics program, t his goal must be a chiev ed — I also
spend quite a b it of time to tell th e reader wh y thing s are pro ved the way they are,
especially in the earlier part of each chapte r . At various points there a re (hopefully)
visible attempts to help one “see” a theorem (either by discu s sing informally the “plan
of attack,” or by providing a “false-proof”) i n addition to confirming i ts validity by
means of a f orm al proof. Moreover, whenever it was possible, I ha ve tried to av oid
the rabb it-out-of-the-hat p roofs, and rather g av e rigo rou s argu ments w h ich “explain”
the situation that is being analyzed. Longer proofs are thus often accompanied by
footnotes that describe the basic ideas in more heuristic terms, reminiscent of ho w
one would “tea ch” t h e proof i n the classroom.
1
Th is way the text is hopefully brought
do wn to a level whic h wou ld be readable for most second or third semester gradu ate

studen ts in economics and advanced undergraduates in mathematics, while it still
preserves the aura o f a serious analysis course. Having said this, ho wever, I should
note that th e exposition g ets less r estraine d towards the e nd of e ach chap ter, an d
the analys is is presented witho ut being over ly pedantic. This goes especially for the
“starred” sections which co ver more ad va nc ed ma ter ial than the rest of the text.
The basic approach is, of cou rse, primarily that of a tex tbook r ath er than a refer-
ence. Yet, t he reader wi ll still find here the careful, yet unprov ed, s tatements of a good
n umber of “d ifficult” t heorems that fit well with the o ver all development . (Exam-
ples. Blu mberg’s Theorem, non-co ntractibilit y of the sphere, Radem acher’s Theorem
on the d ifferen tiability of Lipschitz continuous functions, Motzkin’ s Theore m , Reny’s
Theorem on t h e existence of N as h equilibrium, etc ) A t the very least, this should
hint at the student what migh t be expected at a higher lev el course. Furt hermore,
some of t hese results are w idely used in economic theory, s o i t i s de sirable that the
studen ts begin at this stage developing a precursory understanding of them. To this
end, I discuss a few of t hese “difficult” theorems a t some length , ta lk about their
applications, and at times g ive proofs for special cases. It is worth noting that th e
general exposition relies on a select few of these results.
Last, but not least, it is m y sincere hope that the present treatmen t pro vides
glimpses o f th e strength of “abstract reason ing,” may it come from applied ma th e-
matical analysis or f rom pure analysis. I ha ve tried hard to strike a balance i n t his
regard . Overall, I put far mo re emphasis on the app licability of the main theorems
relative to their generaliza tion s and /or st rong est formulations, only rarely me ntion
if someth in g can be achieved without i nvoking the Axiom of Choice, and use the
method of “proof-by-con trad iction ” more frequently than a “purist” might like to
see. On the other hand, by means of various remarks, exercises and “starred” sec-
1
In keeping with this, I have written most of the footnotes in the first person singular pronoun,
while using exclusively the first person plural pronoun in the body of the text.
xi
tions, I try to touch upon a few topics that carry more of a pure mathematician’s

emphasis. (Examples. The characterization of metric spaces with the Banach fixed
poin t property, th e conv e rse of Weierstrass’ Theorem, various ch ar acterizations of in-
finite dimensional n ormed linear spaces, etc ) This v ery much reflects my full accord
with the following wise w ords of Tom K ö rner:
“ Agoodmathematiciancanlookataprobleminmorethanoneway.
In p articu lar, a good ma the m atic ian will ‘think li ke a pure m at he matician
when doing pure m athematics and like an applied mathematician when
doing applied mathe matics’. (Great mathematicians think like thems elves
when doing mathematics.)”
2
On the Structure of the Text. T h is book consists of four parts:
I. Set Theory (Chapters A-B)
II. Analysis on Metric Spaces (Chapters C-E)
III. Analysis on Linear Spa ces (Chapters F-H)
IV. Analysis on Metric/Normed Linear Spaces (Chapters I-K)
Part I pro v id es an elementary, yet fairly comprehensive, ov erview of (intuitive) set
theory. Cov ering the fundamen tal notions of sets, relations, functions, real sequences,
basic calculus, and countability, this part is a prerequ isite for the rest of th e text.
It also introduces the Axiom of Choice and some of its equivalen t formulations, and
sketches a precursory introduction to o rder theory. A m on g the most notable theorems
covered here are Tarski’s Fixed Poin t Theorem an d S ziplrajn’s Extension Theorem.
Part II is (almost) a standard course on real analy sis on metric spaces. It s tud -
ies at leng th the (topological) properties of separability and compactn es s, and t h e
(uniform) p roperty of completeness, along with the theory of continuous functions
and correspondences, in the context of metric s paces. I also talk about the elemen ts
of fixed point theory ( in E uclidean spaces) and i ntroduce the theories of stationary
dynamic programmin g an d N as h equ ilib rium. Among the m os t no tab le theorems
co vered here are the C ontraction M apping Principle, the Stone-Weierstrass Theorem,
the Tietze Extension Theorem, Berge’s Maxim um T h eorem, the fixed point theorems
of Bro uwer and Kakutani, a nd Mic hael’s Selection T heorem.

Part III begins with an extensiv e review of some line a r algebraic concepts (such as
linear spaces, bases and dimension, linear operators, etc.), and then proceeds to con-
vex analysis. A purely lin ea r algebraic treatment of both t h e analytic and geometric
2
Little is lost in translation if one adapts this quote for economic theorists. You de cide:
“ A good economic theorist can look at a problem in m or e than one way. In particular,
a good economic t heorist will ‘think like a pure theorist w hen doing pure econo mic theory
and like an applied theorist when doing applied theory’. (Great economic theorists think
like themselves when doing economics.)”
xii
forms of the Hahn-Banac h Theorem is given here, along with several economic a p-
plications th at range from individual decisio n t h eo r y t o financial economics. Among
the most notable theorems covered are the Hahn-Banac h Extension Theorem, the
Krein-Rutman Theorem, and t he Dieu do nn é S eparation Theorem.
Part IV can be considered as a primer on geo m et ric linear a n d no nlin ea r analysis.
Since I wish to avoid the consideration of general to pology in th is text, the en tire
discussion is couc hed within metric and/or normed linear spaces. The results on
the extension of linear function als an d the separation by hy perplanes are shar pened
in this con text, an in troduction to infin ite d ime n sional convex ana lys is is outline d ,
and t h e fixed po int t heory dev eloped earlier with in Euclidean spaces is carried into
the realm of norm e d linear spaces. The final chapter considers differen tial calculus
and o p timiz ation in Banach space s, a nd b y way of a n application, p rovides an in tro-
ductory, but rigorous, approach to calculus of variations. Amon g t he most notable
theorems covered here are the Separating Hyperplane Theorem, t he Uniform B ound -
edness Principle, t he G licksberg-Fa n F ixed Poin t Theorem, Sc hauder’s Fixed Point
Theorems, and the Krein-Milman T heorem.
On the Exercises. As in most mathematics textbooks, the exercises pro vided
throughout t he text a re integral to the p resen t exposition, and hence a ppear w ithin
the m ain body o f various sec tions. M any o f them appear f ollow ing t he introduction
of a particularly important concept to make the reader get better acquainted with

that concept. Others are given after a major t h eore m in order to illustrate “ho w to
apply” the associated re s ult or the method of p r oof that yie ld ed it.
Som e of th e exercises look like this:
Exer cise 6.
Such exercises are “mu st do” ones whic h will be us ed in the material that follows
them. Other exercises l ook e it he r like
Exer cise 6.
or
Exer cise 6.
I strongly encourage the r eader to work on those of the form er type, for t hey com-
plement t he exposition at a basic level, even thou gh skipping them wo uld not impair
one’s ability to move on to subsequ ent material. Those of th e latter kind, on the
other hand, aim to p rovide a pra ctice ground for t he st uden ts to i mprove t h eir un-
derstanding of the related topic, and at times suggest directions for further study.
3
Wh ile most of the exercises in this book are quite “doable” — well, with a rea son -
able amount of suffering — some are c h alle ngin g (these are “starre d” ) and s ome are
for th e ve ry best s tud ents (th ese are “d oub le starred.”) Hints and partial solutions
3
While quite a few of these exercises are original, several of them come from the problem sets of
m y teachers Tosun Terzioglu, Peter Lax and Oscar Rothaus.
xiii
are provided for about one-third of them at the en d of the book.
4
All in all — and
this will be ab u nd a ntly clear from early on — t h e guiding philosophy behind this text
strongly subscribes to the view t hat t here is only o ne way o f learning mathematics,
and t hat is learning-by-doing . In h is preface, Chae (1995) uses the fo llowing beautiful
Asian prov erb to drive this point home:
I hear, and I forget;

I see, and I remem ber;
I do, and I understand.
This recipe, I submit, should be tried also by those who wish to have some fun
throughout the f ollowing five hundred some pages.
On Measure Theo r y and Integ ration. This te xt omits the theory of measure
and Lebesgue integration in its e ntirety. These topics are covered in a forthcoming
companion volum e called Probability Theory with Economic Applications.
On Alternative Uses of the Text. This book is intend e d to serv e as a textbook
for a number of d ifferent courses, and also for independent study.
• A second graduate course on m athematics-for-economists. Such a course would
use Chapter A for review, a nd co ver t he first section of Chapter B, along with
pretty muc h all of Chapters C , D and E . This should tak e s omething lik e one
half t o t wo-thirds of a sem ester, depending on how long o ne wishes to spend
on the a p p lication s of dynamic programming and game theory. The remaining
part of th e semester may th en be used to go deeper into a va riety of fields,
such as c on v ex analysis (Chapters F—H and parts of Chapters I a nd J ), in tro-
ductory linear analysis (Chap ters F-J), or introductory nonlinea r analys is and
fixed point th eory (pa rts of the C hapters F, G and I, and Chap ters J-K). Alter-
natively, one m ay alter the focus, and offer a little course in probability theory
whose coverage may n ow be accelerated. (For what’s its worth, this is how I
taught from the text a t NY U for about 6 years with some success.)
• A first graduate course on mathematics-for-economi sts. U nless the math prepa-
ration of th e class is extra ord ina ry, this text would n o t serve well as a pr im ary
textbook for this sor t of a course. How e ver, it may be useful for complemen-
tary reading o n a good num ber of topics that are tradition ally cov e red in a
first math-for-eco n course, especially if the i n struc tor wishes to tou ch upon in-
finite dimensional matters as well. (Exam ples. The earlier parts of Chapters
C-E co mplements the s tan dard coverage of rea l an aly sis with in R
n
, Chapter

C spends quite a bit of time on the Contraction Mapp in g Th eo rem an d its
4
To the Student: Please work on the exercises as hard as you can, before seeking out these hints.
This is for y o ur own good. Believe it or not, you’ll thank me later.
xiv
applications, Chapters E provides an extensive coverage of matters related to
correspondences, Cha pters F-G inv estigate linear spaces, operators, and basic
conv ex analysis, and include numerous separating and supporting hyperplane
theor em s of varying generality.)
• An advanced (under )graduate course on real analysis for mathematics students.
While my topic selection is dictated by the needs of m odern economic theory,
the presen t text is fo remost a mathematics book. It is t herefore duly s uitable
to be used as a textbook for a c ourse o n mathematical ana lysis at the s enio r
undergraduate or first year g rad u ate level. Especially if the instructor wishes to
emphasize the fixed poin t theory and s ome economic applications (regarding,
say, individual decision t h e ory), it may well help organ iz e a full fledged math
cour se.
• Inde pendent study. One of the major objectiv es of this text is to provide the stu-
dent with a glimpse o f what lies behind t he horizon o f the standard mathematics
that is cov ered i n th e first y ear o f m o st gra duat e e conomics p rograms. Good
portions of Chapters G-K, for instance, are relatively advanced, and hence may
be deemed unsuitable for the c ourses mentioned a bove. Yet I have t ruly tried
m y best to be able m ak e these chapters accessible t o the studen t who needs
to le arn the related material but finds it di fficult to follow the stan da rd texts
on l inear and nonlinear f unctional analysis. It may eventually be necessary t o
study m atters from more a dva nced treatmen ts, but the coverage of t his book
ma y perhaps ea se the pain by building a bridge bet ween advanced texts a nd a
standard “math-for-econ” course.
On Related Textbooks. A few w ords about how this text fares with other r elated
textbooks a re in order. It will become abundantly clear early on that my treatmen t

is a good deal m ore advanced than that of the excellent introductory book b y Simon
and Blume (1994) an d of t he slight ly more advanced text by d e la Fuente (1999).
While the topics o f Simon and Blume (1994) are prerequis ites for the p res e nt course,
de la Fuente (1999) dovetails w ith my treatm ent. It is, on the other hand , for the
most p art equally adva nced as the popular treatise by Stokey and Luc as (1989) wh ich
is sometim es t aught as a second math course f or economists. Most of what is assumed
to be known in the latter reference is cov ered here. So, upon finishing the present
course, the studen t (who wishes t o take an in troductory class on t he theory of dy nam ic
program m ing an d discrete stoc hastic system s) would be able to read th is book with
a considerably accelerated pace. Similarly, after the presen t course, the advanced
texts like Mas-Colell (1989), D uffie (1996), and B e cker and Bo y d III (1997) sh ou ld
be within reach.
With in the m ath e m a tics folklore, th is book would be v iewed as a c ontinuation of
a first “m a the matical analysis” course , whic h is usually taught after or along with
xv
“advan c ed calculus.” In that sense, it is more adva n c ed than the expositions of
Rudin (1976), R oss (1980), and Körner (2003), an d is rough ly at th e same level w ith
Kolm ogorov and Fomin (1970), H aaser and Sulliva n (1991 ), and Carothers (2000).
The coverag e of the wid ely popular Royden (1986) and Folland (1999 ) ov erlap qu ite
a bit w ith mine as well, but the level of those books are a bit more advanced. Finally,
a r elated text whic h is exceedingly m ore advanced than the p resent one is that o f
Aliprantis and Border (1999). That book covers an amazing pl ethora of t opics from
functional analysis, and should serve as a useful advanced reference book for an y
studen t of economic theory.
Error s. While I desperately tried to avoid them, a nu mber of erro rs ha ve surely
managed to sneak past me. I can only hope that they are not substan tial. The errors
that I have identified after the publication of the book w ill be posted in my webpage
h ttp ://homepages.nyu.edu/~eo1/books.html. Please do not h esitate to email me of
the on es you find — my e mail address is e fe .ok@ny u.e du.
Ac kn owledg ments. There are many econom ists and m athematicians w h o have con -

tributed significantly to this book. My good s tud ents So p hie Bade, B o
˘
gaçhan Ç elen,
Juan Dubra, Andrei Gomberg, Yu sufcan Masatlioglu, Francesc Ortega, Onur Özgür,
Liz Po tamites, M a her Said , and Hilary Sarneski-H ay e s carefully read substan tial parts
of the manu scrip t and identified a good deal of errors. All the figures in the text are
drawn k in dly, and with pains tak i ng care, by Bo
˘
gaçhan Çelen — I o we a lot to him.
In addition, numerous comments and corrections I receive d from J ose A pesteguia,
Jean-Pierre Benoît, Alberto Bisin, Kim Border, Victor Klee, P eter Lax, Claude
Lemaréc h al, Jing Li, Massimo Marin acci, Tapan Mitra, Louise Nirenberg, Debraj
Ra y, Ennio Stachetti, and Srinivasa Varadhan s haped the s tructure o f the t ext con-
siderably. Especially with Jean-Pierre Benoît and Ennio Stachetti I had long discus-
sions about the final product. Finally, I should ac kno wledge that my mathematical
upbringing, and hence the making of this book, owes m uc h to the v ery m an y d iscus-
sions I had with Tapan Mitr a a t C orn ell by his fam ou s little b lackboard.
At the end of th e day, ho wev er, m y g reatest debt is to my s tudent s who hav e
unduly suffered the preliminary stages o f this text. I can only hope th at I was able
to teac h them as much as they taugh t me.
Efe A. Ok
New York, 2005
xvi
PREREQUISITES
This text is intended primarily for an audience who has taken at least one “mathe-
matics for economists” type course at t he graduate lev el o r a n “advanced calculus”
course with proofs. Conseque ntly, it is assum ed that the reader is familiar with th e
basic m e th ods of calculus, linear a lge bra, and nonlinea r (s tatic) o p timization, which
w ould be covered in such a course. For completeness p urposes, a relativ ely compre-
hensiv e review of the ba sic theory of real sequences, f unctions and ordinary calculus

is provided in Chapter A. In fact, many facts concerning real functions are reproved
later in t he text i n a more g eneral co n t ext. Nevertheless, ha ving a good understanding
of real-to-real functions o ften h e lps g ettin g intuition about thing s in more abstract
settings. Finally, w h ile m ost students come acro ss me tric spaces by the end of th e
first semester of their graduate education in econom ics, I do no t assume a ny prior
knowledge of th is t op ic h e re.
To judge things for y o u r se lf, ch eck if you have s ome “ feeling” for the follo wing
facts:
• Ev ery monotonic sequ ence of real numbers in a closed and bounded i nterval
converges in that interv al.
• Ev ery conca ve function definedonanopenintervaliscontinuousandquasicon-
cave.
• Ev ery d ifferen tiab le fun c tion on R is contin uous, but not conversely.
• Ev ery con tinuous fu n ction defined on a closed and bounded in terval attains its
maximum.
• A set of vectors that s pans R
n
has at least n vectors.
• A linear function defined on R
n
is continuous.
• The (Riemann) integral of ev ery contin u ous func tion definedonaclosedand
bounded i nterval equals a finite number.
• The Fundamental Theorem of Calculus.
• The Mean Value Theorem.
If you hav e certainly seen these results before, a n d if yo u can sketch a quick (in-
formal) argumen t r egarding the validity of about half of them, you are w ell prepared
to read this book. (All of these results, or substantial gener alization s of th em , are
prov ed with in th e t ext .)
The economic applications covered here are foundational for the large part, so

they do not require any sophisticated economic training. Ho wever, if you ha ve tak en
at least one graduate course on microeconomic theory, then y ou would probably
appreciate th e importance of these application s better.
xvii
BASIC CONVENTIONS
• Thefrequentlyusedphrase“ifandonlyif”isoftenabbreviatedinthetextas
“iff.”
• Roughly speaking, I label a major resu lt as a theore m, a result l ess sig nificant
than a theorem, but still of inte re st, as a proposition, a more or less immediate
consequence of a theorem or p roposition as a corol lary,aresultwhosemainutility
deriv e s f rom its aid in deriving a th eorem or proposition as a lemma,andfinally,
certain a uxiliary results as claims, facts or ob servations .
• Throughout this text n stands fo r an arbitrary positiv e integer. This sym bol
will c orre spond almost exclusively to the (algebraic) dimensio n of a Euclidean space,
hence the notation R
n
. If x ∈ R
n
, then it is understood that x
i
is th e r e a l number
that corresponds to the ith coordinate of x, that is, x =(x
1
, , x
n
).
• Iusethenotation⊂ in the stric t sens e. Th at is, implicit in the statem ent
A ⊂ B is that A = B. The “subsethood” relation in the weak sense is d enoted by ⊆.
• T hroughout this text the sym bol  sy mbolizes th e endin g of a partic u lar dis-
cussion, ma y it be an example, observation, or a remark. The symbol  ends a claim

within a p roof of a theor em, proposition, etc., wh ile  e nd s th e p roof itself.
• For any symbols ♣ and ♥, the expressions ♣ := ♥ and ♥ =: ♣ m ean that ♣ is
defined by ♥. (This i s the so-called “Pascal notation.”)
• W h ile the c ha p ters are labeled by Latin letters, the sections and subsection s of
the chapter s a re labeled by positive integers. Consider the following sente nc e :
By Proposition 4 , the c on clus ion of C orolla ry B.1 would be va lid here,
so by using the observatio n noted in Example D.3.[2],wefind that the
solution to the problem me ntioned at the end of Section J.7 exists.
Here Proposition 4 refers to the proposition numbered 4 in the c hap ter that this
sentence is taken from. Corollary B.1 is the Corollary 1 of Chapter B, Example
D.3.[2] refers to pa rt 2 of Example 3 that is given in Chapter D, an d Section J.7 is
the s eventh section o f Chapter J. (The orem. The chapter from which this sentence is
taken cannot be any one of the chapters B, D and J.)
• The r est of the no tation and con ventions t hat I a dopt throughout the text are
explained in Chapter A.
xviii