# The Set of Real-Valued Continuous Functions on a Compact Metric Space X, C(X)

We will soon look at a very important theorem known as The ArzelĂ â€“Ascoli Theorem but we will first need to define an important type of metric space. We first define the sets for which our metric space will be over.

Definition: Let $(X, d)$ be a compact metric space. The Set of Real-Valued Continuous Functions on the Compact Metric Space $X$ is denoted $C(X) = \{ f : X \to \mathbb{R} : f \: \mathrm{is \: continuous} \}$. |

For example, consider the compact metric space $([0, 1], d)$ where $d$ is the usual Euclidean metric defined for all $x, y \in \mathbb{R}$ by $d(x, y) = \mid x - y \mid$. Then:

(1)For example, the functions $f_1(x) = x$, $f_2(x) = x^2$, $f_3(x) = \cos x$ all belong to $C[0, 1]$. However, the function $f_4(x) = \frac{1}{x - \frac{1}{2}}$ does not belong to $C[0, 1]$ since $f_4$ is discontinuous at $\frac{1}{2} \in [0, 1]$.

We will now define an important metric on $C(X)$.

Definition: Let $(X, d)$ be a compact metric space. Define a metric $\rho : C(X) \times C(X) \to [0, \infty)$ on $C(X)$ defined for all $f, g \in C(X)$ by $\displaystyle{\rho (f(x), g(x)) = \max_{x \in X} \{ \mid f(x) - g(x) \mid \}}$ so that $(C(x), \rho)$ is a metric space. |

We will now verify that $\rho$ is indeed a metric. Let $f, g, h \in C(X)$.

Suppose that $\rho(f(x), g(x)) = 0$. Then $\max_{x \in X} {\mid f(x) - g(x) \mid} = 0$. So $\mid f(x) - g(x) \mid = 0$ for all $x \in X$ which implies that $f(x) - g(x) = 0$ and $f(x) = g(x)$ for all $x \in X$. Conversely suppose consider $\rho(f(x), f(x))$. We have that then $\rho (f(x), f(x)) = \max_{x \in X} \{ \mid f(x) - f(x) \mid \} = 0$. Therefore $\rho(f(x), g(x)) = 0$ if and only if $f(x) = g(x)$ for all $x \in X$.

We now show that symmetry holds for $\rho$. Note that:

(2)We lastly show that the triangle inequality holds. We see that:

(3)Therefore $\rho$ is indeed a metric and so $(C(X), \rho)$ is a metric space.