FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

FIVE MAJOR RESULTS IN ANALYSIS
AND TOPOLOGY
Aaron Peterson

Contents
Preface iv
Chapter 1. Preliminaries 1
Set Theory 1
Algebra 3
Topology 5
Chapter 2. The Ascoli-Arzelà Theorem 11
Chapter 3. The Stone-Weierstrass Theorem 15
Chapter 4. The Hahn-Banach Theorem 23
Chapter 5. The Baire Category Theorem 30
Chapter 6. The Stone- ˘ Cech Compactification 34
Bibliography 39
iii

Preface
During the analysis of scientific or cultural problems, there often arise abstract logical
objects which seem to accurately model the system from which the problem derives. The
substance of mathematics consists largely of exploring these objects, obtaining ever deeper
insights into their structure, behavior, and other salient properties. Mathematics, through
the sciences which directly employ it, claims among its successes many of the technological
feats of the last few millennia. In fact, it often provides the language through which the
theories of a particular science are rendered.
Naturally, any mathematical result that illuminates some useful and basic property of an
interesting logical structure will be highly valued and praised throughout the mathematical
and scientific communities; after all, an insight into the logical structure is assumed to be
tantamount to an insight into the real system is it supposed to model.
In this paper, we explore five such momentous results that are among the most funda-
mental and indispensable in analysis and topology. Although only a basic understanding of
analysis and topology is sufficient to understand these results, many of them lie in the nexus
of analysis and topology and are therefore rarely encountered in undergraduate, and even
beginning graduate, coursework.
iv

CHAPTER 1
Preliminaries
In order to ground our later discourse, a brief review of the fundamentals is in order. The
purpose of the present discussion is twofold: first, it will establish the linguistic and symbolic
conventions which we will follow throughout the balance of this exposition. Secondly, and
perhaps more importantly, it will provide (or perhaps refresh) for the reader a set of concepts
and results, the comprehension of which is sufficient to understand the motivation, proof,
and consequences of the five theorems which are the main focus of this work.
The reader is assumed to be familiar with the rudiments of set theory, algebra, and
analysis. However, some of this material will be introduced in this chapter for the reasons
outlined above, as well as material from topology and analysis in metric spaces.
Set Theory
A set is a collection of objects. We will often refer to the set which has no elements.
This set is called the null, void, or empty set, and is represented by the symbol ∅.
We will typically use upper-case letters to denote sets, and lower-case letters to denote
their elements. The statement ‘object x belongs to the set X’ or ‘x is a member of X’ will be
expressed symbolically as x ∈ X. Sometimes, it is more convenient to define a set in terms
of a condition which its members must meet. Let C(x) be such a proposition. We define the
set X of objects x which satisfy C(x) as
X = {x| C(x) } .
Table 1 defines a number of standard sets used throughout this paper.
If X and Y are sets, we say that X is a subset of Y if x ∈ X guarantees that x ∈ Y ,
and in this case we write X ⊂ Y . Two sets X and Y are equal if, and only if, X ⊂ Y and
Y ⊂ X.
1

N = {1, 2, 3,...}
R = {x| x is a real number}
R+ = {x| x ∈ R and x > 0}
SET THEORY 2
R+ = {x| x ∈ R and x ≥ 0}
C = x + iy| x,y ∈ R, i = √ −1
J n = {(x1,...,xn)| xi ∈ J for i = 1,...,n}
Table 1. Definitions of Standard Sets
We define the union (or join) of two sets X and Y , as
X ∪ Y = {x| x ∈ X or x ∈ Y } .
Similarly, the intersection (or meet) of two sets X and Y is defined as
X ∩ Y = {x| x ∈ X and x ∈ Y } .
For an arbitrary family of sets Xα, where α is the member of an arbitrary set J of indices,
we can refer to the union or intersection of the Xα, denoted
and
Xα = {x| There is an α ∈ J so that x ∈ Xα}
α∈J
Xα = {x| x ∈ Xα for every α ∈ J}.
α∈J
We can also construct new sets by removing the elements of one set from another. The
complement of X with respect to Y is defined as
Y − X = {x| x ∈ Y and x /∈ X} .
The (Cartesian) product of two sets X and Y , denoted X × Y , is given by
X × Y = {(x,y)| x ∈ X and y ∈ Y } .

ALGEBRA 3
For instance, {a,b}×{0, 1} = {(a, 0), (a, 1), (b, 0), (b, 1)}. Given any arbitrary set J of indices
α, the product over the sets Xα is given by
Xα = {(xα)α∈J| xα ∈ Xα} .
α∈J
Throughout the rest of our present discussion, assume that X, Y and Z are arbitrary
sets. Define Y X = {(yx)x∈X|yx ∈ Y }. This can be thought of both as the set of all X−tuples
of elements of Y (if X is taken to be a set of indices), or as the set of all functions from X
into Y .
Let f be one such function from X into Y . If B ⊂ Y, define f −1 (B) = {x| f(x) ∈ B}.
The set f −1 (B) is called the pullback of B under f.
If X ⊂ Z, and f ′ : Z → Y is such that f(x) = f ′ (x) for every x ∈ X, then f ′ is said to
be an extension of f to Z. Likewise, f is called the restriction of f ′ to X.
The function f is called injective if, for every distinct x1 and x2 in X, f(x1) and f(x2)
are distinct in Y . We say that f is surjective if, for every y in Y , there is an x in X such
that f(x) = y. A function that is both injective and surjective is said to be bijective. If f is
a bijection, then it induces a unique map f −1 : Y → X since the pullback of any point of Y
is nonempty and contains a single point. f −1 is called the inverse of f.
Often, we will need to discuss the size, or cardinality, of a particular set. Let Sn =
{1, 2,...,n}. If there is a bijective function fn : X → Sn for some n ∈ N, then we say that
X is finite, and write |X| = n. If there is a bijective function fN: X → N, then we say that
X is countable, and write |X| = ℵ0. If no such fn or fN exists, we say that X is uncountable.
The following concept will serve us well in the remainder. A sequence in X is a function
f : N → X. Sequences are often written in set notation as {xn| n ∈ N} or {xn}, where
f(n) = xn. A subsequence of a sequence {xn} is the sequence in X formed by restricting
f : N → X to the countable subset N ∗ = {n1,n2,n3,...} of N, where ni < nj if i < j. The
subsequence formed formed by restricting f to N ∗ is denoted {xni }.
Algebra
A group is a set G, together with a binary operation + such that if x,y,z ∈ G:

(1) x + y ∈ G,
(2) (x + y) + z = x + (y + z),
ALGEBRA 4
(3) there is an element 0 ∈ G such that x + 0 = 0 + x = x, and
(4) there is an element −x ∈ G such that (−x) + x = x + (−x) = 0.
The group (G, +) is said to be abelian if the operation + is commutative. That is, if
x + y = y + x for every x,y ∈ G.
A ring is an abelian group (G, +) together with a second binary operation • such that if
x,y,z ∈ G:
(1) (x • y) • z = x • (y • z), and
(2) x • (y + z) = x • y + x • z and (x + y) • z = x • z + y • z.
Note that the operation • is not required to be commutative in the ring (G, +, •), and that
there need not be a multiplicative identity element or multiplicative inverses. If the operation
• is commutative, then (G, +, •) is called a commutative ring. If (G, +, •) is a commutative
ring with a multiplicative identity element 1 (which behaves with respect to • in exactly
the same manner that 0 behaves with respect to +), then (G, +, •) is said to have unity. A
commutative ring with unity in which every non-zero element has a multiplicative inverse is
called a field. We will frequently encounter the fields R and C.
A linear space X over a field A, is an abelian group such that, for every α,β ∈ A, and
for every x,y ∈ X, the following relations hold:
(1) α · x ∈ X
(2) 1 · x = x
(3) α · x + β · x = (α + β) · x
(4) α · x + α · y = α · (x + y)
If A is R or C, then X is said to be a real or complex linear space, respectively. If X is a
linear space over the field A, and Y is a linear space over the field B, with A ⊂ B, then if
f : X → Y satisfies f(α · x + β · y) = α · f(x) + β · y for every x,y ∈ X and α,β ∈ A, f is
said to be linear. If g: X → A, then g is called a functional.

X.
TOPOLOGY 5
Topology
Throughout this section, let X be a set and let ℘(X) denote the class of all subsets of
A topology on X is a subset T of ℘(X) such that:
(1) X and ∅ are elements of T .
n
(2) If U1,...,Un ∈ T , then Ui ∈ T .
i=1
(3) If {Uα} α∈J is a subset of T , then
α∈J
Uα ∈ T .
The set X taken together with the topology T on X is said to be a topological space,
and is sometimes denoted (X, T ). The sets of T are said to be open in X. A subset U of a
topological space X is called Gδ if it can be written as a countable intersection of open sets.
A subset U of X is said to be closed in X if, and only if, X − U is open in X.
Most of the spaces we will study in this paper admit some notion of ‘distance’ between
the elements, and this notion provides critical insights into the structure of the space. We
will, therefore, develop many of our subsequent results with those spaces in mind.
A metric on a set X is a function ρ: X × X → R+ such that for any x,y,z ∈ X,
(1) ρ(x,y) = 0 if, and only if, x = y,
(2) ρ(x,y) = ρ(y,x), and
(3) ρ(x,y) ≤ ρ(x,z) + ρ(z,y).
Metrics are functions which describe the ‘distance’ between any two elements in the space,
and they behave exactly how we think ‘distances’ ought.
If X is a topological space that has a metric ρ defined on it, then for x ∈ X and any
ǫ > 0, we define the ǫ-ball centered at x as Bǫ(x) = {y| ρ(x,y) < ǫ}. We say that the metric
ρ induces the topology of X if Bǫ(x) is open in X for each x ∈ X and ǫ > 0, and if, for any
open set U of X containing the point x ∈ X, there is an ǫ > 0 such that Bǫ(x) ⊂ U. If the
metric ρ induces the topology of X, then (X,ρ) is said to be a metric space.
Let f : X → Y . Then f is said to be continuous if, for each open set U of Y , the pullback
of U under f is open in X. If (X,ρX) and (Y,ρY ) are metric spaces, then this definition

TOPOLOGY 6
implies that f is continuous in the topological sense if, and only if, for each ǫ > 0 one can
find a δ > 0 such that ρX(x,y) < δ ensures that ρY (f(x),f(y)) < ǫ.
If A is a subset of the topological space X, define the map j : A → X by j(a) = a. The
map j is continuous, and is called the inclusion map from A to X. The function i: A → A
defined by i(a) = a is continuous and is called the identity map on A. In general, we will say
that, for topological spaces X and Y , C(X,Y ) will denote the set of all continuous functions
from X into Y .
A function f : X → Y is called open if the image of every open set of X is open in Y .
Similarly, a function g: X → Y is called closed if the image of every closed set of X is closed
in Y .
A bijective continuous function from X into Y that has a continuous inverse is called a
homeomorphism between X and Y . Since a homeomorphism induces a bijection between
the topologies of X and Y , a homeomorphism preserves any topological structures present.
Hence, two spaces that are homeomorphic can be viewed, topologically, as one and the same.
A sequence {xn} in a metric space X is called fundamental if, for every ǫ > 0, one can
find an N ∈ N such that for m,n ≥ N, ρ(xm,xn) < ǫ. That is, a fundamental sequence is
one in which the terms eventually get close together.
Closely related to fundamentality is the concept of convergence. A sequence {xn} in a
metric space X is said to converge if there is a point x ∈ X such that, for every ǫ > 0, there
is an N ∈ N such that if n > N, then ρ(x,xn) < ǫ. If this is the case, we say that x is the
limit of the sequence {xn}, and write lim
n→∞ xn = x. Since the limit of a convergent sequence
is unique, then any subsequence of a convergent sequence will converge to the same limit as
the original sequence. A metric space X is said to be complete if every fundamental sequence
in X converges.
We have, then, a useful characterization of continuous functions between metric spaces,
which is stated in the following theorem.
Theorem. Let f : X → Y , where (X,ρX) and (Y,ρY ) are metric spaces. Then a
necessary and sufficient condition for f to be continuous is that for every x ∈ X and sequence
{xn} converging to x, {f(xn)} converges to f(x).

TOPOLOGY 7
Proof. (Necessity) Let x ∈ X, and {xn} be a sequence converging to x. Given ǫ > 0,
choose δ > 0 such that if y ∈ Bδ(x), then f(y) ∈ Bǫ (f(x)). Choose an N ∈ N such that for
every n > N, xn ∈ Bδ(x). Then f(xn) ∈ Bǫ (f(x)).
(Sufficiency) Suppose that f is not continuous at a point x ∈ X. Let δn = 1
. Then for
n
some ǫ > 0, choose for each n ∈ N a xn such that xn ∈ Bδn(x), but f(xn) /∈ Bǫ (f(x)). Then
{xn} converges to x, but {f(xn)} does not converge to f(x), contrary to hypothesis.
If A is a subset of the metric space Y , the the set of points {yα} α∈J is said to be ǫ-dense
with respect to A if, for each a in A, one can find an α ∈ J such that ρ(a,yα) < ǫ.
If U is a subset of a topological space X, then we say that x ∈ X is a limit point of U
if every open set of X containing x also contains a point y of U. If X is a metric space,
then this definition is equivalent to saying that x is a limit point of U if, and only if, there
is a sequence {un} in U which converges to x in X. The closure A of a subset A of the
topological space X is the set A together with the limit points of A. A can also be described
as the smallest closed set containing A.
A subset U of a topological space X is dense in X if U = X. The following theorem will
prove useful in our later discussion.
in V .
Theorem. U is dense in X if, and only if, for every open set V of X, U ∩ V is dense
Proof. Suppose U is dense in X, and that V is an open subset of X, Then U ∩ V ⊂
U ∩ V = X ∩ V = V . If x ∈ V , and if O is an open set containing x, then there is a point
v ∈ O such that v ∈ V . Since O ∩ V is open, and v ∈ O ∩ V , then the density of U in X
guarantees that there is a point u ∈ O ∩ V ⊂ O, such that u ∈ U. Hence, V ⊂ U ∩ V , so
that U ∩ V = V . The converse follows immediately by noting that X is an open subset of
itself.
One of the most important and useful properties that a topological space can possess is
called compactness. Let A be a subset of the topological space X, and {Uα} α∈J be a class of
open sets in X. If A ⊂
α∈J
Uα, we say that {Uα} α∈J is an open covering of A. If J ∗ ⊂ J, and

A ⊂
β∈J ∗
TOPOLOGY 8
Uβ, then {Uβ} β∈J ∗ is said to be a subcover of A. The set A is said to be compact
if, and only if, every open covering of Y contains a finite subcover of Y .
Compactness will appear at several crucial junctures throughout our discussion, and there
are several equivalent definitions of it which we will apply. The following theorem outlines
those definitions.
Theorem. Let X be a metric space, and let A be a subset of X. Then the following are
equivalent:
(1) A is compact in X.
(2) Every set in X that is ǫ-dense with respect to A admits a finite set that is also
ǫ-dense with respect to A.
(3) Every sequence in A contains a subsequence which converges.
Proof. (1 → 2) Let ǫ > 0. Consider the open cover {Bǫ(a)} a∈A of A. Since A is
compact, there is a finite subcover {Bǫ(ai)} i∈Sn of A. Then a1,...,an is the desired ǫ-dense
set.
(2 → 3) Let {bn} be a sequence in A. Define ǫn = 1
n for each n ∈ N. Let A1 be a finite
ǫ1-dense set in A, and choose a1 ∈ A1 such that there is an infinite number of points of {bn}
in Bǫ1(a1). We proceed as follows: given an−1, let An be a finite ǫn-dense set in An−1, and
choose an ∈ An such that there are an infinite number of points of {bn} in An−1 ∩ Bǫn(an).
Continue ad infinitum. Now, for each n ∈ N, choose a sequence of points bni
such that
bni ∈ Bǫi (ai). Then {bni } is fundamental. Since each bni ∈ Bǫ1 ∩ A, and Bǫ1 ∩ A is closed in
the subspace A, we have that {bni } converges in A.
(3 → 1) 1 This leg of the proof requires two lemmas, which we now prove.
Lemma. For every open covering {Uα} of A, there is a δ > 0 such that for each x ∈ X
and 0 < ǫ < δ, there is an α ∈ J such that Bǫ(x) ∈ Uα.
Proof. Suppose that the contrary is true, and chose, for each n ∈ N, a set Cn such that
Cn ⊂ B 1 (x) for some x, but there is no α such that Cn ⊂ Uα. Let {xn} be a sequence in X
n
1 This proof was adapted from [8] and [13].

TOPOLOGY 9
such that xn ∈ Cn. By hypothesis, {xn} contains a subsequence {xni } which converges to a
point x ∈ A. Choose α such that x ∈ Uα. Since Uα is open in the metric space X, choose
ǫ > 0 so that Bǫ(x) ⊂ Uα, and choose i so large that 1
ni
< ǫ
2 . Then Cni ⊂ Bǫ(x) ⊂ Uα,
which contradicts our supposition.
Lemma. Given ǫ > 0, one can find a finite ǫ-dense subset of A.
Proof. Suppose, on the contrary, that there is an ǫ > 0 such that no finite ǫ-dense set of
X exists. Choose x1 ∈ X. By hypothesis there exists an x2 ∈ X such that x2 /∈ Bǫ(x1). Given
n
x1,...,xn, choose xn+1 such that xn+1 /∈ Bǫ(xi). Since ρ(xn,xj) ≥ ǫ for j = 1,...,n − 1,
i=1
{xn} cannot have a convergent subsequence.
We now conclude the proof of the theorem. Let {Uα} α∈J be an open cover of A. By our
first lemma, we can choose an a > 0 such that for every x ∈ A and δ < a, Bδ(x) ⊂ Uα for
some α ∈ J. Let ǫ = a
3 . By our second lemma, choose a finite ǫ-dense set x1,...,xn in A.
For each 1 ≤ j ≤ n, choose Uj such that Bǫ(xj) ⊂ Uj. Then {Uj} n
j=1
is a finite subcover of
A.
We proceed by developing a bit more machinery involving mappings into metric spaces.
A function f from the space X into the metric space Y is bounded if, for some point y ∈ Y ,
one can find a positive real number M such that f(X) ⊂ BM(y).
If A is a set of bounded functions from the space X into the metric space Y , define the
metric ρ∞(f,g) = supρY
(f(x),g(x)) on A. We call ρ∞ the uniform metric on A. If {fn} is
x
a sequence of functions from X into Y , and {fi(x)} converges for each x ∈ X, then we say
that {fn} converges. Define f : X → Y by f(x) = lim
n→∞ fn(x). We call f the pointwise limit
of {fn}.
Let {fn} be a sequence of functions from X into the metric space Y with pointwise limit
f. If {fn} is fundamental with respect to ρ∞, then {fn} is said to converge to f uniformly.
It is well known that the pointwise limit of a uniformly convergent sequence of continuous
functions is itself continuous.
Let (X,ρX) and (Y,ρY ) be metric spaces. Then f : X → Y is said to be uniformly
continuous if, for every ǫ > 0, one can find a δ > 0 such that for every x ∈ X, x0 ∈ Bδ(x)

TOPOLOGY 10
ensures that f(x0) ∈ Bǫ (f(x)). We will sometimes consider sets of functions which share a
certain ‘degree’ of continuity. A set G of functions from X into Y is equicontinuous if, given
ǫ > 0, one can a δ > 0 such that if x1,x2 ∈ X and ρX(x1,x2) < δ , then ρY (f(x1),f(x2)) < ǫ
for every f ∈ G.
Example. For α > 0, let Fα be the class of all real-valued functions f on the closed unit
interval satisfying the condition |f(x) − f(y)| ≤ α|x − y|, where x,y ∈ [0, 1]. Then f ∈ Fα
is uniformly continuous, since given ǫ > 0, one must merely select δ < ǫ
α
to ensure that
|x − y| < δ guarantees that |f(x) − f(y)| < ǫ. Furthermore, since δ does not depend on f,
we see that Fα is equicontinuous.

CHAPTER 2
The Ascoli-Arzelà Theorem
One often finds that significant mathematical results which establish the existence of
some object in a space often rest on crucial properties of the space in question, such as
compactness. Naturally, any theorem which establishes a set of necessary and sufficient
conditions for compactness of a space could spawn any number of existential results. The
Ascoli-Arzelà theorem is one such example. It claims that a subset D in the space of
continuous functions from one compact metric space to another is compact if, and only if,
its members are equicontinuous.
The proof of the Ascoli-Arzelà theorem will be abbreviated by the following result.
Theorem (Heine-Cantor). 1 Let (X,ρX) be a compact metric space, and (Y,ρY ) a metric
space. Then every f ∈ C(X,Y ) is uniformly continuous.
Proof. Suppose, on the contrary, that there existed an f0 ∈ C(X,Y ) and an ǫ0 > 0 such
that for every δ > 0, one can find x,y ∈ X so that ρX(x,y) < δ, but ρY (f0(x),f0(y)) ≥ ǫ0.
For n ∈ N, choose xn,yn so that ρX(xn,yn) < 1
n , but ρY (f0(xn),f0(yn)) ≥ ǫ0. Since
X is compact, one can find a convergent subsequence {xnk } of {xn}. Let x = lim xnk
k→∞ .
Similarly, choose a convergent subsequence {ynk } of {yn}, and set y = lim ynk . Since the
k→∞
sequence xn1,yn1,xn2,yn2,... is fundamental in a compact space, let g be its limit. Since every
subsequence of a convergent sequence converges to the same limit, we have x = lim
k→∞ xnk =
g = lim
k→∞ ynk
= y. Then ρX(xnk ,ynk ) < 1
nk and ρY (f0(xnk ),f0(ynk )) ≥ ǫ0 for every k ∈ N.
This contradicts our assumption that f0 ∈ C(X,Y ).
1 The proof of this theorem is similar to that found in [12]
11

2. THE ASCOLI-ARZELÀ THEOREM 12
Theorem (Ascoli-Arzelà). 2 Let (X,ρX), (Y,ρY ) be compact metric spaces, and let D ⊂
C(X,Y ). Then a necessary and sufficient condition that D be compact in C(X,Y ) is that D
be equicontinuous.
Proof. (Sufficiency) Let Y X = {f| f : X → Y }, and give Y X the uniform metric ρ∞.
Since convergence with respect to ρ∞ is equivalent to uniform convergence, and the point-
wise limit of a uniformly convergent sequence of continuous functions is again continuous,
C(X,Y ) is closed in Y X . Hence it is sufficient to show that D is compact in Y X , since the
intersection of a compact subspace and a closed subspace is again compact.
Given ǫ > 0, choose δ > 0 so that for every function f ∈ D, ρX(x1,x2) < δ implies that
-dense in X. Let
ρY (f(x1),f(x2)) < ǫ. Since X is compact, let {xi} n
i=1 be δ
2
and for 1 < i ≤ n, set
H1 = B δ(x1),
2
Hj.
i−1
Hi = B δ(xi)
−
2
j=1
The Hi form a pairwise disjoint cover of X, such that if x1,x2 ∈ Hi then ρX(x1,x2) < δ
2 .
Similarly, the compactness of Y allows us to pick a finite ǫ
m
-dense set {yj} 2 j=1 .
Let G be the set of all functions from X to Y which assume, on each Hi, the value of
exactly one yj. It is clear that |G| = m n < ∞. It remains to be shown that G is ǫ-dense in
Y X with respect to D.
Let f ∈ D. For each 1 ≤ i ≤ n, choose yj so that ρY (f(xi),yj) < ǫ/2, and choose g ∈ G
so that g(xi) = yj. Given x ∈ X, choose Hi so that x ∈ Hi. Then
ρY (f(x),g(x)) ≤ ρY (f(x),f(xi)) + ρY (f(xi),g(xi)) + ρY (g(xi),g(x)) < ǫ ǫ
+ + 0 = ǫ.
2 2
(Necessity) Let ǫ > 0. Since D is compact, we can choose a finite ǫ-dense
subset G of
3
D. The Heine-Cantor theorem allows us to select, for each g ∈ G, a constant δg > 0 so that
if ρX(x1,x2) < δg, then ρY (g(x1),g(x2)) < ǫ
3 . Let δ = ming {δg}.
2 The sufficiency portion of this proof is adapted from [6]

2. THE ASCOLI-ARZELÀ THEOREM 13
Hence, if f ∈ D, then a g ∈ G can be chosen so that ρ∞(f,g) < ǫ.
It follows that, for
3
x1,x2 which satisfy ρX(x1,x2) < δ,
ρY (f(x1),f(x2)) ≤ ρY (f(x1),g(x1)) + ρY (g(x1),g(x2)) + ρY (g(x2),f(x2)) < ǫ.
Its appearance in the basic results of analysis makes the Ascoli-Arzelà theorem particu-
larly interesting and powerful. For this reason, the applications of the Ascoli-Arzelà theorem
are many and varied, extending to a number of diverse branches of mathematics. We now
present several important results which follow from it. These examples, taken from geom-
etry, complex analysis, and the theory of differential equations, hint at the scope of the
Ascoli-Arzelà theorem.
A curve in a metric space X from a point a to a point b is a continuous function f :
[0, 1] → X such that f(0) = a and f(1) = b. The length of f is defined by
n
sup
0=x0

2. THE ASCOLI-ARZELÀ THEOREM 14
neighborhood D of z and a constant M so that every f ∈ X G , the modulus of f is bounded
by M within D.
Theorem (Montel 4 ). Let H(G) denote the collection of all analytic functions on a fixed
region G. Then a necessary and sufficient condition for F ⊂ H(G) to be normal is that F
be locally bounded.
The Ascoli-Arzelà theorem is used, not surprisingly, to prove sufficiency.
The next result extends the reach of the Ascoli-Arzelà theorem into the realm of differ-
ential equations.
Theorem (Peano 5 ). Let y ′ (x) = f (x,y(x)) be a given differential equation. If f (x,y(x))
is continuous in a closed region G containing a point x, then there is a solution to the given
equation which passes through x.
Peano’s existence theorem for ordinary differential equations is one of the most basic
theorems in the field. Its proof involves constructing a sequence of polygonal arcs through
x which, by the Ascoli-Arzelà theorem, converges to a continuous solution of the differential
equation through x.
4 Adapted from [3]
5 Adapted from [6]

CHAPTER 3
The Stone-Weierstrass Theorem
One of the most important structures studied in mathematical analysis is the set C(X,Y )
of all continuous functions from a compact space X (such as [0, 1]) into a metric space Y
(such as R or C). An understanding of this set is crucial for the development of modern
and classical calculus, differential equation theory, differential geometry, and several other
branches of mathematics. For many applications in classical analysis, X is taken to be a
closed interval of the real line, and Y is taken to be either C or, as a special case, R. One
particular subset of C([0, 1], C) - the class of all complex polynomials on [0, 1] - is particularly
easy to study. Karl Weierstrass showed that, in the metric space (C([0, 1], C),ρ∞), for any
function f and any ǫ > 0, one can find a polynomial p such that ρ∞(f,p) < ǫ. This result is
known as the Weierstrass approximation theorem. There are two widely known constructive
proofs of this theorem. The version presented here uses a set of polynomials known as
the Landau kernels, and follows that given in [12]. The other proof uses the Bernstein
polynomials, and can be found in [13].
Theorem (Weierstrass). Let f be a continuous complex-valued function on [a,b]. Then
there exists a sequence of polynomials Pn such that Pn(x) → f(x) uniformly on [a,b]. If f
is real, then the Pn can be taken to be real.
Proof. Without loss of generality, assume that [a,b] = [0, 1], and that f(0) = f(1) = 0.
Furthermore, define f(x) to be zero for every x /∈ [0, 1]. Then f is uniformly continuous on
the entire real line. Let Qn(x) = cn(1 − x 2 ) n for each n ∈ N, where cn is chosen to ensure
15

that 1
−1 cn(1 − x 2 ) n dx = 1. Since
3. THE STONE-WEIERSTRASS THEOREM 16
1
−1 (1 − x2 ) n dx = 2 1
0 (1 − x2 ) n dx
≥ 2 1/ √ n
(1 − x 0
2 ) ndx ∗
≥ 2 1/ √ n
(1 − nx 0
2 )dx
= 4
3 √ n
> 1
√ n ,
then cn < √ n. To justify step ∗, note that (1 − x 2 ) n − (1 − nx 2 ) = 0 if x = 0, and
d 2 n 2 2 n−1 (1 − x ) − (1 − nx ) = n(1 − x ) (−2x) + 2nx
dx
= 2nx (1 − (1 − x2 ) n−1 ) > 0 (x ∈ (0, 1)).
Hence, for any 0 < δ ≤ 1, Qn(x) ≤ √ n(1 − δ 2 ) n (δ ≤ |x| ≤ 1), so that Qn → 0 uniformly
for all x such that δ ≤ |x| ≤ 1. Now set Pn(x) = 1
−1 f(x + t)Qn(t)dt (0 ≤ x ≤ 1). Because
[0, 1] ⊂ [x − 1,x + 1] for x ∈ [0, 1], our assumptions about f guarantee that
Pn(x) = x+1
x−1 f(t)Qn(t − x)dt
= 0
x−1 f(t)Qn(t − x)dt + 1
0 f(t)Qn(t − x)dt + x+1
1 f(t)Qn(t − x)dt
= 0 + 1
0 f(t)Qn(t − x)dt + 0,
which is clearly a polynomial in x. Thus, {Pn} is a sequence of polynomials (which are real
if f is real). Since f is a continuous on a compact set, for ǫ > 0 choose δ > 0 such that
|y − x| < δ guarantees that |f(y) − f(x)| < ǫ
. Let M = sup |f(x)|. Since Qn(x) ≥ 0, then
2 x
for 0 ≤ x ≤ 1,
1
|Pn(x) − f(x)| =
[f(x + t) − f(x)] Qn(t)dt
−1 1
≤ |f(x + t) − f(x)|Qn(t)dt
−1 −δ
≤ 2M Qn(t)dt + ǫ
δ
Qn(t)dt + 2M
2
−1
≤ 4M √ n(1 − δ 2 ) n + ǫ
2
< ǫ
−δ
1
δ
Qn(t)dt
for sufficiently large n. This completes the proof.

3. THE STONE-WEIERSTRASS THEOREM 17
The Weierstrass approximation theorem allows us to choose a sequence of polynomials
which converge uniformly to a given continuous function f. Since the pointwise limit of a
uniformly convergent sequence of functions has many of the same properties as the terms in
the sequence, this theorem tells us a great deal about C([a,b], C).
There is a significant generalization of Weierstrass’ approximation theorem due to Mar-
shall Stone, which relaxes two of the conditions of Weierstrass’ theorem. First, set of func-
tions is not required to be polynomials, but rather a type of subclass A of C(K, C), called an
algebra, which possesses a few important properties. Also, K is not required to be the closed
interval [a,b], but rather is allowed to be any compact topological space. This result is known
as the Stone-Weierstrass theorem, and its proof is accomplished in two main steps. First,
we prove a special case of the theorem for the space C(K, R) which relies on the classical
Weierstrass approximation theorem. We then use this result to prove the theorem in its full
strength. Before we begin, a few preliminary results are in order.
Definition. Let K be compact, A be a scalar field, and let A be a linear subspace over
A of C(K,A) such that for f,g ∈ A, f · g ∈ A, where (f · g)(x) = f(x) · g(x). Then A is
called an algebra of continuous functions. A is said to be real or complex if A is R or C.
An algebra of continuous functions is merely a ring of continuous functions, inheriting
addition and multiplication from A, and which is closed under scalar multiplication.
Definition. Let A be a subset of the metric space (C(K,A),ρ∞), where A is R or C.
Then the closure of A in C(K,A) is called the uniform closure of A, and A is said to be
uniformly closed in C(K,A).
The uniform closure of an algebra A can be thought of as the set of all functions in
C(K,A) which are the limit of a fundamental sequence in A.
Theorem. Let A be an algebra of bounded functions. Then the uniform closure B of A
is a uniformly closed algebra.
Proof. For f ∈ B, g ∈ B, let {fn}, {gn} be uniformly convergent sequences in A such
that fn → f, gn → g. Since the fi and gi are bounded, fn + gn → f + g, fngn → fg,

3. THE STONE-WEIERSTRASS THEOREM 18
and cfn → cf uniformly, where c is a constant. Hence, B is an algebra. Let {bn} be a
fundamental sequence of functions in B, and for each n ∈ N let {fn,i} be a sequence of
functions in A which converge uniformly to bn. Then {fn,n} is a fundamental sequence in
A, and hence there is a function F ∈ B such that fn,n → F, so that bn → F. This shows
that B is uniformly closed.
Definition. Let A be a subset of C(K,A), where A is R or C. A is said to separate
points on K if, for each x1,x2 ∈ K, one can find an f ∈ A such that f(x1) = f(x2). A
is said to vanish at no point of K if, for each x ∈ K, there is a function f ∈ A such that
f(x) = 0.
Lemma (1). Suppose A is an algebra of functions on a set E such that A separates
points on E and A vanishes at no point of E. Then if x1,x2 ∈ E are distinct, and c1,c2 are
constants (real if A is a real algebra), then there is a function f ∈ A such that f(x1) = c1,
and f(x2) = c2.
Then
Proof. Let g,h,k ∈ A be chosen such that
g(x1) = g(x2), h(x1) = 0, k(x2) = 0.
f(x) = c1 (g(x) − g(x2)) · h(x)
(g(x1) − g(x2)) · h(x1) + c2 (g(x1) − g(x)) · k(x)
(g(x1) − g(x2)) · k(x2)
satisfies the conclusion of the lemma.
We now arrive at the first generalization of Weierstrass’ approximation theorem.
Theorem (Stone-Weierstrass Theorem (Real Version)). Let A be a real algebra of con-
tinuous functions on a compact set K. If A separates points on K and if A vanishes at no
point of K, then the uniform closure B of A is C(K, R)
The Stone-Weierstrass theorem for spaces of real continuous functions with compact
domain will be proved as a sequence of four lemmas. The proof is adapted from that found
in [12].

3. THE STONE-WEIERSTRASS THEOREM 19
Lemma (Part 1). If f ∈ B, then |f| ∈ B, where |f|(x) = |f(x)| for every x ∈ K.
Proof. Let a = sup |f(x)| and ǫ > 0. By the Weierstrass approximation theorem,
x
choose a sequence of polynomials {P ∗ n(x)} such that P ∗ n(x) → |x| uniformly on [−a,a]. Define
Pn(x) = P ∗ n(x)−P ∗ n(0). Then one can find real numbers c1,...,cn such that
ǫ for every −a ≤ y ≤ a. Since B is an algebra, g =
n
|y|
− ciy
i=1
i
<
n
cif i is in B, so that |g(x) − |f(x)|| < ǫ
for each x ∈ K. Since B is uniformly closed, |f| ∈ B.
Before proceeding, we must define the maximum and minimum of two functions.
Definition. If f and g are real functions defined on a space X, then the maximum of f
and g is a real function max(f,g) defined on X such that max(f,g)(x) = max (f(x),g(x))
for each x ∈ X. The function min(f,g) is defined analogously. If f1,...fn are real functions
defined on a space X, then we define max(f1,...,fn) recursively by
min(f1,...,fn) is defined analogously.
i=1
max (...max (max(f1,f2),f3),...,fn).
Lemma (Part 2). If f1,...,fn ∈ B, then max(f1,...,fn),min(f1,...,fn) ∈ B.
Proof. If f,g ∈ B, then since
max(f,g) =
min(f,g) =
f + g
2
f + g
2
|f − g|
+ and
2
|f − g|
− ,
2
the fact that max(f,g) ∈ B and min(f,g) ∈ B is a consequence of Part 1. The conclusion
immediately follows via induction and the preceding definition.
Lemma (Part 3). If f ∈ C(K, R) and ǫ > 0, then for each point x of K there is a function
gx in B such that gx(x) = f(x), and gx(t) > f(t) − ǫ for every t ∈ K.
Proof. Choose x ∈ K. Since A ⊂ B, A separates points of K and vanishes nowhere on
K, then Lemma 1 guarantees, for each y ∈ K, the existence of a function hy ∈ B such that
hy(x) = f(x) and hy(y) = f(y). The continuity of hy allows us to choose an open set Jy of

3. THE STONE-WEIERSTRASS THEOREM 20
K containing y such that hy(t) > f(t) − ǫ. Since K is compact, we can choose a finite set of
n
points y1,...,yn such that K ⊂ Jyi . Set gx = max (hy1,...,hyn). Part 2 guarantees that
i=1
gx ∈ B. By construction, gx(t) > f(t) − ǫ for every t ∈ K, and gx(x) = f(x).
Lemma (Part 4). If f ∈ C(K, R) and ǫ > 0, then there is a function h in B such that
|h(x) − f(x)| < ǫ for every x ∈ K.
Proof. For each point x in K, construct gx as in Part 3. Since gx is continuous,
choose, for each point x in K, an open set Vx of K containing x such that gx(t) < f(t) + ǫ
m
for every t ∈ K. Since K is compact, choose x1,...,xm such that K ⊂ Vxi . Put
h = min (gx1,...,gxm). By Step 2, h ∈ B. The construction process of h guarantees that
f(t) − ǫ < h(t) < f(t) + ǫ for every t ∈ K, which is what was to be shown.
The complex Stone-Weierstrass theorem introduces one more requirement on our subal-
gebra A, namely that it is self-adjoint.
Definition. We say that a subset A of C(K, C) is self-adjoint if, for each f ∈ A, f ∈ A,
where f(x) = f(x).
Since every subset of C(K, R) is self adjoint, this condition could have been added to the
premises of the last theorem. Hence, the following result includes the preceding theorem as
a special case.
Theorem (Stone-Weierstrass Theorem (Complex Version)). If A is a complex self-
adjoint algebra of continuous functions on a compact set K such that A separates points
of K and vanishes nowhere on K, then the uniform closure B of A is C(K, C).
Proof. Let AR be the set of all real continuous functions on K. We note that AR ⊂ A.
If f ∈ A and f = u+iv (u,v : K → R), then 2u = f +f, and since A is a self-adjoint algebra,
it follows that u ∈ AR. Choose distinct points x1,x2 of K. Since A separates points, there
is a function f in A such that f(x1) = 1 and f(x2) = 0. Hence, 1 = u(x1) = u(x2) = 0, so
that AR separates points on K. If x is a point of K, then since A vanishes nowhere on K
we are guaranteed the existence of a g ∈ A and a complex scalar λ such that λg(x) > 0. If
i=1

3. THE STONE-WEIERSTRASS THEOREM 21
f = λg and f = u + iv, then u(x) > 0, so that AR vanishes at no point of K. By the real
version of the Stone-Weierstrass theorem, the uniform closure of AR in A is C(K, R), and is
contained therefore in B, the uniform closure of A. The result follows from noting that B is
a complex algebra, and that if f ∈ C(K, C) then f = u + iv is in B because u,v are in the
uniform closure of AR ⊂ B.
There is an even further generalization of the Stone-Weierstrass theorem, which was
proved by Erret Bishop in 1961. The result, Bishop’s Theorem, can be found in [1] and [11].
As discussed before, the Stone-Weierstrass theorem and its special cases appear through-
out analysis, but primarily in results that involve maps into scalar fields. We will conclude
by providing a few interesting results which rely on this theorem.
The following extension of the Weierstrass approximation theorem which, although it is a
trivial corollary of the Stone-Weierstrass theorem, can be used to extend many of the results
of one-dimensional analysis into two dimensions.
Theorem (Extension of Weierstrass Approximation Theorem 1 ). If K ⊂ R 2 is compact,
and if f ∈ C(K, C), then there is a polynomial g(x1,x2) such that |f(x1,x2) − g(x1,x2)| < ǫ
for every (x1,x2) ∈ K.
The following result, from functional analysis, in incredibly useful. The n th moment of a
real continuous function f defined on [0, 1] is the value of the integral
1
0
f(x)x n dx.
Theorem (Moments of Real Functions 2 ). Let f,g be real continuous functions defined
on [0, 1]. Then f and g are identical if, and only if, the n th moments of f and g agree for
every n ∈ N.
This last result implies the following theorem.
Theorem (Number of Continuous Real Functions). The cardinality of the set of contin-
uous real functions defined on [0, 1] is equivalent to that of R.
1 Taken from [13].
2 Taken from [13].

1
3. THE STONE-WEIERSTRASS THEOREM 22
Proof. The previous theorem implies the map F : C([0, 1], R) → Rω given by F(f) =
f(x)x n
dx is injective, so that if |N| = ℵ0, and |R| = ℵ1, then |C([0, 1], R)| ≤ |R
n∈N
ω | =
ℵ0 ℵ0×ℵ0 ℵ0 = 2 = 2 = ℵ1 = |R|. To show that |R| ≤ |C([0, 1], R)|, we note that
0
ℵ1 ℵ0 = 2 ℵ0
the map g: R → C([0, 1], R) given by t ↦→ ft, where ft(x) ≡ t, is injective.
Interestingly enough, there is at least one other proof of the preceding theorem which
also relies on the Weierstrass approximation theorem. In this alternate case, it is shown that
every real polynomial can be approximated by polynomials with rational coefficients, and
hence that the set of all real polynomials on [0, 1] with rational coefficients can approximate
any continuous function on [0, 1] arbitrarily well. The result then follows from the cardinality
of C([0, 1], R) must be less than or equal to the cardinality of the power set of the set of all
polynomials with rational coefficients, which, since there are countably such polynomials,
is equal to the cardinality of the continuum. The proof that the cardinality of C([0, 1], R)
is greater than or equal to the cardinality of the continuum is accomplished in exactly the
same way as in the above theorem.

CHAPTER 4
The Hahn-Banach Theorem
Functional analysis encompasses, not surprisingly, the study of linear spaces and the set
of mappings whose domain is a linear space (including functionals). Three results are abso-
lutely necessary for the development of this field: the Open Mapping and Banach-Steinhaus
theorems which will be discussed in the next chapter, and the Hahn-Banach theorem, which
will be the focus of our current discussion. Before we can state the theorem, a number of
concepts and results need to be developed.
X,
Definition. Let X be a set. If ≤ is a binary relation on X such that, for every x,y,z ∈
(1) x ≤ x,
(2) x ≤ y and y ≤ x implies that x = y, and
(3) x ≤ y and y ≤ z implies that x ≤ z,
then ≤ is called a partial ordering of X. X, when given the relation ≤, is called a partially
ordered set. A chain in X is a sequence of elements x1,...,xn,... ∈ X such that x1 ≤ · · · ≤
xn ≤ · · ·.
Definition. If X is a partially ordered set, then x is called a maximal element in X if,
for each y ∈ X, x ≤ y if, and only if, x = y.
Axiom (Zorn’s Lemma). Let P be a partially ordered set in which every chain has an
upper bound. Then P possesses a maximal element.
23

4. THE HAHN-BANACH THEOREM 24
This controversial axiom is equivalent to the axiom of choice and the principle of well
ordering. It does, however, furnish a simple proof 1 of the Hahn-Banach theorem, and we will
therefore employ it.
Definition. Let A and B be linear subspaces of the linear space X. Then the linear
subspace of X spanned by A and B is the set of all sums of the form a+b, where a ∈ A and
b ∈ B. If A is the set of all scalar multiples of a single element a of X, then A is denoted
[a].
that
Definition. Let X be a linear space. A norm on X is a function · : X → R+ such
(1) x = 0 if, and only if, x = 0,
(2) α · x = |α| · x, and
(3) x + y ≤ x + y.
If X has a norm · , then (X, · ) is called a normed linear space.
Theorem. Every normed linear space is a metric space.
Proof. The function ρ(x,y) = x − y is a metric. This follows immediately from the
definition of a norm.
Hence, we talk about the completeness of normed linear space in the same sense as the
completeness of a metric space. A complete, normed linear space is called a Banach space.
The norm · induces more than a metric on the linear space X; it also induces a norm
on the set of all functionals on X.
1 The proof we follow here is adapted from [4] and [13]. Another proof of the theorem which uses Zorn’s
lemma, but is restricted to spaces of countable dimension is found in [9]. For a proof that utilizes the axiom
of choice, see [6].

4. THE HAHN-BANACH THEOREM 25
Definition. Let f be a functional on the normed linear space X, and suppose that there
is a c ≥ 0 such that for any x1,x2 ∈ X, |f(x1) − f(x2)| ≤ c · x1 − x2. Then the norm of
f, f, is defined by
f = inf c.
Then f is said to be bounded, and it follows that f(x1) − f(x2) ≤ f · x1 − x2.
If X is a linear space, we will use the symbol X ∗ to denote the set of all bounded linear
functionals on X.
We can also use the norm · to define continuity of functionals in much the same way
that we defined continuity of functions between metric spaces.
Definition. A functional f : X → A is continuous at a point x if, for every sequence
{xn} in X converging to x, we have limn f(xn) − f(x) = 0. If f is continuous at every x
in X, then f is said to be continuous.
Boundedness and continuity of functionals are crucially intertwined, as the following
theorem shows.
Theorem. A necessary and sufficient condition for a linear functional f to be continuous
is that f be bounded. 2
Proof. Suppose f is bounded. Then f(xn) − f(x) ≤ f · xn − x, so that
limnf(xn) − f(x) = 0 if limnxn − x = 0, and hence f is continuous.
Conversely, suppose f is continuous. Then f is continuous at 0. Thus, choose a 0 <
δ < 1 so that y < δ implies that f(y) ≤ 1. Then, if x = 0, we have f(x) =
2x
δ · f
δx
2x
≤ 2x
δ , so that f is bounded.
We now introduce the Hahn-Banach theorem, which states that any bounded linear
functional defined on a linear subspace M of the normed linear space N can be extended
to a bounded linear functional on N without increasing its norm. This implies that any
continuous linear functional defined on a subspace of a normed linear space extends to a
continuous functional defined on the entire space.
2 Both this theorem and its proof can be found in [7].

4. THE HAHN-BANACH THEOREM 26
The proof will be completed in two parts. First, we prove a lemma which allows us to
extend any linear functional defined on a linear subspace M of the normed linear space N
in an additional single dimension, while preserving the norm of the functional. We then
conclude with a brief reducio ad absurdum argument that is equivalent to the conclusion of
the theorem.
Lemma (Extension of Linear Functionals). Let N be a normed linear space, M a linear
subspace of N, and f ∈ M ∗ . If y ∈ N − M, let M0 = M + [y] be the linear subspace of N
spanned by M and y. Then f can be extended to a functional f0 ∈ M ∗ 0 so that f0 = f.
Proof. First we prove the theorem in the case where N is a real normed linear space.
We will use this result to establish the theorem for complex normed linear spaces.
Suppose, without loss of generality, that f = 1. Since y /∈ M, every v ∈ M0 can be
written uniquely as v = x + αy, where x ∈ M and α ∈ R. If f0 is a linear extension of f to
M0, then f0(v) = f(x) + αf0(y). Hence, it we are done when we find a value for f0(y) that
ensures that f0 = 1. This equivalent to the condition that
(0.1) |f0(x + αy)| ≤ x + αy
for every x ∈ M,α = 0. Since f0(x + αy) = f(x) + αf0(y), we use (0.1) to write
(0.2) −f
Now, we note that if x1,x2 ∈ M, then
so that
x
−
α
x
α + y ≤ f0(y) ≤ −f
x
+
α
x
+ y.
α
f(x2) − f(x1) ≤ fx2 − x1 = x2 − x1 ≤ x2 + y + x1 + y,
(0.3) −f(x1) − x1 + y ≤ −f(x2) + x2 + y.
Define a = sup {−f(x) − x + y} , and b = inf {−f(x) + x + y}. By (0.3), a ≤ b, so we
x
x
need merely choose f0(y) ∈ [a,b] to complete the proof for real normed linear spaces.
Now, suppose that N is a complex normed linear space. Let f be a complex functional
on M with f = 1. Then f(x) = g(x) + ih(x), where g and h are real linear functionals on
M. Since f = 1, g ≤ 1.

4. THE HAHN-BANACH THEOREM 27
Note that since f(ix) = if(x), f(ix) = g(ix) + ih(ix) and if(x) = ig(x) − h(x), h(x) =
−g(ix) so that f(x) = g(x)−ig(ix). Since g is a real linear functional on M, we can extend g
to a real functional g0 on M0 in such a way that g0 = g. Define f0(x) = g0(x)−ig0(ix) for
all x ∈ M0. Certainly f0 is an extension of f from M to M0, and f0(αx+βy) = αf0(x)+βf0(y)
for all α,β ∈ R. This last equality holds for complex α,β as well, since
f0(ix) = g0(ix) − ig0(i 2 x) = i (g0(x) − ig0(ix)) = if0(x),
so that f0 is a complex linear functional on the complex space M0.
It remains to be shown that f0 = 1. Clearly, f0 ≥ f = 1. Let x ∈ M0 such that
x = 1. If f0(x) ∈ R, then f0(x) = g0(x) and g0 ≤ 1, so that |f0(x)| ≤ 1. If f0(x) ∈ C,
then f0(x) = reiθ , r > 0, so |f0(x)| = r = e−iθ
−iθ −iθ f0(x) = f0 e x . Since e x = x = 1,
the proof is complete.
Definition. Let {fα} α∈J be a set of functions defined on subsets {Aα} α∈J of a set X,
such that fα = fβ in Aα ∩ Aβ for each α,β ∈ J. The the union of the fα is a function f
defined on
Aα such that f agrees with fα on Aα for each α ∈ J.
α∈J
Theorem (Hahn-Banach). Let L be a normed linear space, and M a linear subspace
of L. Then every bounded linear functional f on M has an extension F on L such that
F = f.
Proof. Let f ∈ M ∗ be bounded, and let G be the class of all complex functionals which
extend f to a linear subspace of L containing M, where g ∈ G if, and only if, g = f.
Define on G the relation ≤, where g1 ≤ g2 if g2 is an extension of g1. This relation is a
partial ordering on the set of all linear extensions of f with the same norm as f. Since the
union of every chain of functionals in G is a functional on L which agrees with each of the
functionals in the chain on their domains, and the domain of each functional is bounded
by L, then every chain in G is bounded above. By Zorn’s lemma, there exists a maximal
extension F ∈ G of f. If the domain of F is not L, then we could extend F via our previous
theorem, contradicting the fact that F was maximal.

4. THE HAHN-BANACH THEOREM 28
We conclude with a few interesting results which follow from the Hahn-Banach theorem.
The first two, the theorems of Runge and Müntz-Szasz, are generalizations of the Weier-
strass approximation theorem. The first, by Runge, involves the approximation of analytic
functions in compact subsets of the complex plane by rational functions whose singularities
are contained in some open subset of the plane.
Theorem (Runge 3 ). Let G be an open set in the plane, S 2 be the completed plane, A be
a set which has one point in each maximal connected subset of S 2 − G, and assume that f
is a analytic function on G. Then there exists a sequence {Rn} of rational functions, with
singularities only in A, such that Rn converges uniformly to f on compact subsets of G.
If G is taken to be C, then Runge’s theorem states that, for any entire (analytic every-
where) function f, one can find a sequence of complex polynomials Pn that converges to f
on any compact subsets of C, a clear and useful extension of Weierstrass’ theorem.
A novel generalization of the approximation theorem by Herman Müntz and Otto Szasz
involves the type of polynomials which can be used in the approximation of real-valued
continuous functions.
Theorem (Müntz-Szasz 4 ). Suppose 0 < λ1 < λ2 < λ3 < ... and let X be the closure in
C ([0, 1], R) of the set of all finite linear combinations of the functions 1,t λ1 ,t λ2 ,.... Then
X = C ([0, 1], R) if, and only if,
∞
i=1
1
λi
= ∞.
More specifically, if we want to use linear combinations of the terms t λ1 ,t λ2 ,... to
approximate continuous real functions on the closed interval [0, 1], how must the λi be
distributed in R+? When the λi are taken to be the positive integers, we get the Weierstrass
approximation theorem (for real-valued functions).
The next theorem describes a sufficient condition for a space to be isomorphic to, or have
the same structure as, C
3 Taken from [10]
4 Taken from [10].

4. THE HAHN-BANACH THEOREM 29
Definition. A Banach Algebra X over a field A is a Banach space with an additional
associative operation, multiplication, such that for any x,y ∈ X, x · y ≤ x · y.
Theorem (Gelfand-Mazur 5 ). Every complex Banach algebra with unity in which each
nonzero element has an inverse is isomorphic to C
This is particularly interesting, since it shows that C is, in some sense, the only complete
normed complex linear space that is also an algebraic field.
5 Taken from [10].

CHAPTER 5
The Baire Category Theorem
The Baire category theorem is one of the most elementary results in analysis; its proof is
simple and it relies on no major intermediate results, unlike the Hahn-Banach or the Stone-
Weierstrass theorems. Its significance lies in the insight it provides into the topological
structure of complete metric spaces, which can be easily exploited to prove many powerful
results.
Recall that a subset U of the topological space X is said to be dense in X if U = X, or
if for every open set V of X, U ∩ V = V . In contrast, U is said to be nowhere dense in X if
there is no non-empty open set V of X such that U ∩ V is dense in V .
We pause to prove two lemmas.
Lemma. A is nowhere dense in (X,ρ) if, and only if, A is nowhere dense in (X,ρ).
Proof. Suppose first that U is a nonempty open subset of X, A is nowhere dense, and
that A ∩ U = U. Since A ∩ U ⊂ U, A ∩ U ⊂ U. Let u ∈ U, and choose a sequence {xn} in
A ∩U which converges to u. Since xn ∈ A ∩U choose, for each n, a sequence {xn,i} in A ∩U
such that lim
i→∞ xn,i = xn. Then {xn,n} is a sequence in A ∩ U which converges to u, so that
u ∈ A ∩ U, and hence A ∩ U = U. This contradicts our assumption that A was nowhere
dense.
Now suppose C is nowhere dense in X, B ⊂ C, and U is a nonempty open subset of X.
Suppose that B ∩ U = U. Since B ∩U ⊂ C ∩U ⊂ U, B ∩ U ⊂ C ∩ U ⊂ U ⊂ B ∩ U, so that
C ∩ U = U, contradicting the assumption that C was nowhere dense. We have shown that a
subset of a nowhere dense set is nowhere dense. The proof is complete when we notice that
A ⊂ A.
30

5. THE BAIRE CATEGORY THEOREM 31
Lemma. If X − A is closed and nowhere dense in X, then A is open and dense in X.
Proof. Certainly A is open, being the complement of a closed set. Suppose that U is
a nonempty open subset of X such that A ∩ U is not dense in U. Let V = U − A ∩ U. If
V were empty, then U ⊂ A ∩ U and hence U ⊂ A ∩ U, and since A ∩ U ⊂ U, A ∩ U ⊂ U,
which contradicts our assumption that A ∩ U is not dense in U. Thus, V is nonempty.
Then V = U ∩ (X − A ∩ U) is open, and it follows that V ⊂ A ∩ (X − U) ⊂ A. But then
V ⊂ A ∩ V and A ∩ V ⊂ V , which contradicts our assumption that A was nowhere dense.
This completes the proof.
The term ‘category’ with respect to a space refers to its ability to be written as a ‘small’
union of ‘small’ subsets of itself. A space that can be written as the countable union of
nowhere dense subsets of itself is said to be of the first category. Any set that is not of the
first category is said to be of the second category. The following theorem, by René-Louis
Baire, characterizes non-empty complete metric spaces along these lines.
Theorem (Baire Category Theorem 1 ). Let X be a complete metric space, and let {An}
be a countable collection of dense open subsets of X. Then
n An is dense in X.
Proof. Let Br0(x0) be an open ball in X of radius r0 centered at x0 ∈ X. Since A1 is
open, choose x1 ∈ A1∩Br0(x0) and 0 < r1 < 1 such that Br1(x1) ⊂ A1∩Br0(x0). Given xn−1,
we proceed by choosing a point xn ∈ X and 0 < rn < 1
n such that Brn(xn) ⊂ An∩Brn−1(xn−1).
Since xm ∈ Brn(xn) if m ≥ n, {xi} is fundamental, so that the completeness of X guarantees
the existence of a point x ∈ X such that lim
n→∞ xn = x. Since x lies in each of the closed
spheres Brn(xn), x ∈ Br0(x0)
n An, which is what was to be shown.
The following form of the Baire category theorem is equivalent to the one already proved,
but it casts the result in terms of categories.
1 We follow here the same proof provided in [5] and [2].

5. THE BAIRE CATEGORY THEOREM 32
Corollary. No complete metric space is of the first category.
Proof. Let {An} be a sequence of nowhere-dense subsets of a complete metric space
X. Then
X − An is a countable collection of dense open sets of X. By the preceding
theorem, let x ∈
n (X − An). Then x ∈ X −
n An, so that X =
n An.
The implications of the Baire category theorem are found wherever complete metric
spaces are studied. Considering that the theorem applies to R, C and arbitrary Banach
spaces, this includes most of analysis. One familiar result follows immediately from the
corollary above. Although its proof is not as flashy as the one provided by G. Cantor it is,
however, significantly shorter.
Theorem. R is uncountable.
Proof. Let Ax = {c} for each x ∈ R. Each Ax is nowhere dense, since if y ∈ R − Ax,
then B|x−y|(y) is a neighborhood of y which is disjoint from Ax. If R were countable, then
R would be the countable union of nowhere-dense sets, so that R is not a complete metric
space, which is absurd.
The following two theorems were mentioned at the beginning of the section on the Hahn-
Banach theorem as essential results in functional analysis. They follow immediately from
the Baire category theorem. The first, the Banach-Steinhaus theorem, implies that either
the set M of all bounded linear functions from a Banach space X into a normed linear space
Y is itself bounded in the space of all linear functions from X into Y , or that the set of
all functionals fx from X into R given by fx(y) = f(x), where f ranges through M, is
unbounded in M for all x in some dense subset of X. In other words, either the set of all
bounded linear functions from X to Y is nice everywhere, or it is nasty almost everywhere.
Theorem (Banach-Steinhaus Theorem 2 ). Suppose X is a Banach space, Y is a normed
linear space, and {∆α} is a collection of bounded linear functions from X into Y , where α
ranges over some index set A. Then either there exists an M < ∞ such that ∆α ≤ M for
every α ∈ A, or sup α ∆αx = ∞ for all x belonging to some dense Gδ subset of X.
2 Taken from [10].

5. THE BAIRE CATEGORY THEOREM 33
The second result is called the Open Mapping theorem. It uses linearity to show that it
is not only the case that surjective bounded linear transformations are continuous, but that
they are open maps.
Definition. Let X be a normed linear space. Then the unit ball in X, B1(0), is the set
of all elements of X whose norm is less than unity.
Theorem (The Open Mapping Theorem 3 ). Let U and V be the open unit balls of the
Banach spaces X and Y , respectively. To every bounded linear transformation ∆ of X onto
Y , there corresponds a δ > 0 so that δV ⊂ ∆(U). That is, every surjective bounded linear
transformation of one Banach space onto another is an open map.
3 Taken from [10].

CHAPTER 6
The Stone-˘Cech Compactification
If X and Y are topological spaces, and h : X → Y is a continuous function such that
h is a homeomorphism between X and h(X), then h is called an imbedding of X in the
topological space Y . Often, topological qualities of a space X can be more easily discerned
by imbedding it in a ‘nice’ space Y and analyzing X as a subspace of Y .
Many spaces studied in topology are such that for distinct points x1,x2 one can find
disjoint open sets U1 and U2 containing x1 and x2, respectively. Any space that possesses
this quality is said to be Hausdorff. If Y is a compact Hausdorff space, and X is a dense
subset of Y , then Y is said to be a compactification of X. Since spaces that are compact
and Hausdorff are relatively easy to study, compactifications are a valuable tool in the study
of topological spaces.
If X is a Hausdorff space, it may be the case that for any closed set A and point x ∈ X −A
there is a continuous real function f on X such that f(x) = 0 and f(a) = 1 for every a ∈ A.
Any space X with with this property is said to be completely regular. Every completely
regular space is Hausdorff, and many spaces frequently encountered in analysis and topology
are completely regular.
Definition. A space X is said to be locally compact if, for every point x ∈ X, one can
find an open subset U of X containing x such that U has a compact closure.
If X is a completely regular space, one can construct a compactification of X by adding
one point, called ∞, to X. In fact, this can be done if X is locally compact Hausdorff, but
any completely regular space is locally compact Hausdorff. This is, in a sense, the smallest
compactification of X, in that we add only a single point to obtain a compact Hausdorff
space containing X. There are other ways to compactify a space, but one begins to wonder
34

6. THE STONE- ˘ CECH COMPACTIFICATION 35
if there is a ‘biggest’ compactification of the completely regular space X. It turns out that
such a compactification exists, and it was discovered by Marshall Stone and Eduard ˘ Cech.
Theorem (Stone- ˘ Cech Compactification 1 ). Let X be completely regular. Then there
exists a unique compactification β(X) of X such that
(1) Every continuous map of X into a compact Hausdorff space Y extends uniquely to
a continuous mapping of β(X) → Y , and
(2) Every compactification Y of X is equivalent to a quotient space of β(X).
The proof of this theorem is quite large, and will be shown by a sequence of lemmas. We
will introduce the theory of quotient spaces when it is needed.
Definition. Two compactifications Y1 and Y2 of a space X are said to be equivalent if
there is a homeomorphism from Y1 to Y2 that fixes X.
Lemma (1). Let X be a space, and let h be an imbedding of X into the compact Hausdorff
space Z. Then h induces a unique compactification Y of X, such that there is an imbedding
H of Y into Z which agrees with h on X.
Proof. Given h, let X0 = h(X) ⊆ Z, and let Y0 be the closure of h(X) in Z. Then Y0
is a compact Hausdorff space and equals the closure of X0, so Y0 is a compactification of X0.
Choose a set A that corresponds bijectively with Y0 − X0 via the map k: A → Y0 − X0. Let
Y = X ∪ A, and define H : Y → Y0 by H(x) = h(x) for x ∈ X, and H(a) = k(a) for a ∈ A.
H is bijective because h and k are bijections. Let U ⊂ Y be open in Y if, and only if, H(U)
is open in Y0. Then H is automatically a homeomorphism, being a bijective continuous map.
X is a subspace of Y because H agrees with the homeomorphism h when restricted to X.
Hence, H is an imbedding of Y into Z.
Now, suppose Y1, Y2 are two compactifications of X and that H1 : Y1 → Z and H2 :
Y2 → Z are imbeddings that agree with h on X. Then H1(X) = H2(X) = h(X) = X0.
Furthermore, since H1 and H2 are continuous and Y1 and Y2 are closed, H1(Y1) = H2(Y2) =
1 The statements and proofs of lemmas 1 through 5 are taken almost directly from [8]. An alternate
proof for lemma 6 can be found in [5], although the one provided here is much simpler.

6. THE STONE- ˘ CECH COMPACTIFICATION 36
Y0. Then H −1
2 ◦ H1 is a homeomorphism of Y1 with Y2 that fixes X, so that Y1 and Y2 are
equivalent.
The following lemma is taken from [8], and its proof will be omitted.
Lemma (2). Let X be a space in which one-point sets are closed. Suppose that {fα} α∈J
is an indexed family of continuous functions fα : X → R such that for each x0 ∈ X and for
each open set U of X containing x0, there is an α ∈ J such that fα is positive at x0 and
vanishes outside of U. Then F : X → R J , defined by F(x) = (fα(x)) α∈J is an imbedding of
X in R J . If fα maps X into [0, 1] for each α, then F is an imbedding of X into [0, 1] J .
We construct the Stone- ˘ Cech compactification of X, β(X), as follows: Let {fα} α∈J be
the collection of all bounded continuous real functions on X. For each {fα} α∈J , choose a
closed interval Iα in R containing fα(X). For instance, let Iα = [inf fα(x), sup fα(x)]. Let
x x
h : X →
Iα be defined by h(x) = (fα(x)) α∈J . By the Tychonoff theorem, which states
α∈J
that the arbitrary product of compact spaces is compact,
α∈J
Iα is compact. Because X is
completely regular, {fα} separates points from closed sets in X, so that h is an embedding
by Lemma 2. Let β(X) be the unique compactification of X induced by the imbedding h
guaranteed by Lemma 1. Then there is an imbedding Hβ of β(X) into
Iα that agrees
with h when restricted to X.
Definition. A real function f is said to be bounded if there exists a non-negative real
number M such that |f(x)| ≤ M for each x in X.
Lemma (3). Let X be a space. Then every bounded continuous map f : X → R extends
uniquely to a continuous map of β(X) into R.
Proof. If f ∈ {fα} α∈J , then f = fβ for some β. Let πβ :
α∈J
α∈J
Iα → Iβ be defined by
πβ ((xα)α∈J) = xβ. πβ is clearly continuous, so that πβ ◦ Hβ : β(X) → Iβ is a continuous
extension of f to β(X), since for x ∈ X, πβ (Hβ(x)) = πβ (h(x)) = πβ (fα(x)) α∈J = fβ(x).
By Lemma 1, πβ ◦ Hβ is unique.
The next lemma proves Part (1) of the theorem.

6. THE STONE- ˘ CECH COMPACTIFICATION 37
Lemma (4). Every continuous map of X into a compact Hausdorff space Y extends
uniquely to a continuous mapping of β(X) into Y .
Proof. Since Y is completely regular, it can be imbedded in [0, 1] J for some J by
Lemma 2, so we can assume that Y ⊂ [0, 1] J . Then each component function fα of f is a
bounded continuous real function of X, so by Lemma 3, fα extends uniquely to a continuous
map gα of β(X) into R. Let g : β(X) → R J be defined by g(x) = (gα(x)) α∈J . Then g
is continuous because R J has the product topology, and from this continuity we note that
g(β(X)) = g X ⊂ g(X) = f(X) ⊂ Y = Y , so that g maps β(X) into Y .
Lemma 5 shows the uniqueness of the Stone- ˘ Cech compactification.
Lemma (5). Any two compactification of X with property (1) are equivalent.
Proof. Let β1(X) and β2(X) be two compactifications of X satisfying the extension
property (1) of the Stone- ˘ Cech compactification. Let j2: X → β2(X) be the inclusion map.
Since j2 is continuous, and because β1(X) has property (1), there is a unique continuous
extension f2 of j1 to β1(X). Similarly, the inclusion map j1 : X → β1(X) has a unique
continuous extension f1 defined on β2(X). Then f1 ◦ f2 maps β1 into itself and fixes X, so
that f1 ◦ f2 is a continuous extension of iX, the identity map of X. Similarly, f2 ◦ f1 is a
continuous extension of iX. By Lemma 4, f1 ◦ f2 = iβ1(X), and f2 ◦ f1 = iβ2(X), so that f1
and f2 are homeomorphisms which fix X.
Definition. Let X and Y be topological spaces. Then a surjective map f : X → Y is
said to be a quotient map if a set U is open in Y if, and only if, f −1 (U) is open in X.
Lemma. If X is Hausdorff and A is a compact subset of X, then A is closed.
Proof. We will show that X − A is open. Given a point x ∈ X − A, choose, for each
a ∈ A, disjoint open sets Ua and Va containing x and a, respectively. Since {Va} a∈A is an
open cover of A, there is a finite subcollection of these open sets, say Va1,...,Van covering
n
A. Then U = is open, x ∈ U, and U ∩ A is empty.
i=1
Uai

6. THE STONE- ˘ CECH COMPACTIFICATION 38
If f : X → Y is a quotient map, define X ∗ = {f −1 (y)| y ∈ Y }. The partition X ∗ of X, is
called a quotient space of X.
Lemma (6). Every compactification Y of X is equivalent to a quotient space of β(X).
Proof. We construct a continuous, surjective, closed map g: β(X) → Y which fixes X.
Let j : X → Y be the inclusion map. Since j is continuous and Y is compact Hausdorff, j
extends uniquely to a continuous map j ′ on β(X) via Lemma 4. If U is a closed set in β(X)
then U is compact in β(X). By the continuity of j ′ it follows that j ′ (U) is compact in Y .
By the previous lemma, j ′ (U) is closed in Y . Since Y is the smallest closed set containing
X, and X ⊂ j ′ (β(X)), it must be the case that j ′ (β(X)) = Y . We have shown that j ′ is a
quotient map. The result follows immediately.
This last lemma demonstrates what is meant by saying that the Stone- ˘ Cech compacti-
fication is the ‘biggest’ compactification of a completely regular Hausdorff space X; every
compactification of X can be mapped bijectively into a quotient space of β(X).
The Stone- ˘ Cech compactification is not directly useful in the same way or degree that the
theorems of Ascoli-Arzelà, Stone-Weierstrass, Hahn-Banach, and Baire are. Whereas these
other theorems primarily provide powerful insights about the structures of a space X, the
Stone- ˘ Cech compactification characterized the set of all compactifications of X as a subset
of the class of all quotient spaces of β(X).

Bibliography
[1] Errett Bishop. A Generalization of the Stone-Weierstrass Theorem. Pacific Journal of Mathematics,
11(3):777–783, 1961.
[2] Errett Bishop. Foundations of Constructive Analysis. McGraw-Hill Book Company, Inc., New York,
1967.
[3] John B. Conway. Functions of One Complex Variable. Springer-Verlag, New York, second edition, 1978.
[4] Bernard Epstein. Linear Functional Analysis: An Introduction to Lebesgue Integration and Infinite
Dimensional Problems. W. B. Saunders Company, Philadelphia, 1970.
[5] John L. Kelley. General Topology. D. Van Nostrand Company, Inc., Princeton, 1955.
[6] A. N. Kolmogorov and S. V . Fomin. Elements of the Theory of Functions and Functional Analysis,
volume 1. Graylock Press, Rochester, N.Y., 1957. Trans. Leo F. Boron.
[7] Peter Linz. Theoretical Numerical Analysis. Dover Publications, 1979 edition, 2001.
[8] James R. Munkres. Topology. Prentice Hall, Upper Saddle River, New Jersey, second edition, 2000.
[9] Frigyes Riesz and Bela Sz.-Nagy. Functional Analysis. Frederick Ungar Publishing Co., New York, 1955.
Trans. Leo F. Boron.
[10] Walter Rudin. Real and Complex Analysis. McGraw-Hill Book Company, New York, 1966.
[11] Walter Rudin. Functional Analysis. McGraw-Hill Book Company, New York, 1973.
[12] Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill Book Company, New York, third
edition, 1976.
[13] George F. Simmons. Introduction to Topology and Modern Analysis. McGraw-Hill Book Company, Inc.,
New York, 1963.
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