The Set of R-V Cts. Functions on a Compact Metric Space X, C(X)

# 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 Arzelà–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)
\begin{align} \quad C[0, 1] = \{ f : [0, 1] \to \mathbb{R} : f \: \mathrm{is \: continuous} \} \end{align}

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)
\begin{align} \quad \rho(f(x), g(x)) = \max_{x \in X} \{ \mid f(x) - g(x) \mid \} = \max_{x \in X} \{ \mid g(x) - f(x) \mid \} = \rho (g(x), f(X)) \end{align}

We lastly show that the triangle inequality holds. We see that:

(3)
\begin{align} \quad \rho(f(x), h(x)) &= \max_{x \in X} \{ \mid f(x) - h(x) \mid \} \\ \quad &= \max_{x \in X} \{ \mid f(x) - g(x) + g(x) - h(x) \mid \} \\ \quad & \leq \max_{x \in X} \{ \mid f(x) - g(x) \mid + \mid g(x) - h(x) \mid \} \\ \quad & \leq \max_{x \in X} \{ \mid f(x) - g(x) \mid \} + \max_{x \in X} \{ \mid g(x) - h(x) \mid \} \\ \quad & \leq \rho (f(x), g(x)) + \rho(g(x), h(x)) \end{align}

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