Space of continuous functions on a compact space

In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space

X

{\displaystyle X}

with values in the real or complex numbers. This space, denoted by

C

(

X

)

,

{\displaystyle {\mathcal {C}}(X),}

is a vector space with respect to the pointwise addition of functions and scalar multiplication by constants. It is, moreover, a normed space with norm defined by

‖

f

‖

=

sup

x

∈

X

|

f

(

x

)

|

,

{\displaystyle \|f\|=\sup _{x\in X}|f(x)|,}

the uniform norm. The uniform norm defines the topology of uniform convergence of functions on

X

.

{\displaystyle X.}

The space

C

(

X

)

{\displaystyle {\mathcal {C}}(X)}

is a Banach algebra with respect to this norm.(Rudin 1991, §10.3(a))

Properties

By Urysohn's lemma,

C

(

X

)

{\displaystyle {\mathcal {C}}(X)}

separates points of

X

{\displaystyle X}

: If

x

,

y

∈

X

{\displaystyle x,y\in X}

are distinct points, then there is an

f

∈

C

(

X

)

{\displaystyle f\in {\mathcal {C}}(X)}

such that

f

(

x

)

≠

f

(

y

)

.

{\displaystyle f(x)\neq f(y).}

The space

C

(

X

)

{\displaystyle {\mathcal {C}}(X)}

is infinite-dimensional whenever

X

{\displaystyle X}

is an infinite space (since it separates points). Hence, in particular, it is generally not locally compact.

The Riesz–Markov–Kakutani representation theorem gives a characterization of the continuous dual space of

C

(

X

)

.

{\displaystyle {\mathcal {C}}(X).}

Specifically, this dual space is the space of Radon measures on

X

{\displaystyle X}

(regular Borel measures), denoted by

rca

⁡

(

X

)

.

{\displaystyle \operatorname {rca} (X).}

This space, with the norm given by the total variation of a measure, is also a Banach space belonging to the class of ba spaces.