3The topology of C(K)
II Linear Analysis
3 The topology of C(K)
Before we start the chapter, it helps to understand the title. In particular, what
is
C
(
K
)? In the chapter,
K
will denote a compact Hausdorff topological space.
We will first define what it means for a space to be Hausdorff.
Definition
(Hausdorff space)
.
A topological space
X
is Hausdorff if for all
distinct
p, q ∈ X
, there are
U
p
, U
q
⊆ X
that are open subsets of
X
such that
p ∈ U
p
, q ∈ U
q
and U
p
∩ U
q
= ∅.
Example. Every metric space is Hausdorff.
What we want to look at here is compact Hausdorff spaces.
Example. [0, 1] is a compact Hausdorff space.
Notation. C(K) is the set of continuous functions f : K → R with the norm
kfk
C(K)
= sup
x∈K
|f(x)|.
There are three themes we will discuss
(i)
There are many functions in
C
(
K
). For example, we will show that given
a finite set of points
{p
i
}
n
i=1
⊆ K
and
{y
i
}
n
i=1
⊆ R
, there is some
f ∈ C
(
k
)
such that
f
(
p
i
) =
y
i
. We will prove this later. Note that this is trivial for
C
([0
,
1]), since we can use piecewise linear functions. However, this is not
easy to prove if
K
is a general compact Hausdorff space. In fact, we can
prove a much stronger statement, known as the Tietze-Urysohn theorem.
(ii)
Elements of
C
(
K
) can be approximated by nice functions. This should be
thought of as a generalization of the Weierstrass approximation theorem,
which states that polynomials are dense in
C
([0
,
1]), i.e. every continu-
ous function can be approximated uniformly to arbitrary accuracy by
polynomials.
(iii)
Compact subsets of
C
(
K
). One question we would like to understand is
that given a sequence of functions
{f
k
}
∞
n=1
⊆ C
(
K
), when can we extract
a convergent subsequence?