Boundedness of a Subset of C(X)

# Boundedness of a Subset of C(X)

Recall from The Set of Real-Valued Continuous Functions on a Compact Metric Space X, C(X) page that if $(X, d)$ is a compact metric space the set $C(X)$ is defined to be the set of all real-valued continuous functions on $X$, that is:

(1)
\begin{align} \quad C(X) = \{ f : X \to \mathbb{R} : f \: \mathrm{is \: continuous.} \} \end{align}

We then defined a very important metric $\rho : C(X) \times C(X) \to [0, \infty)$ given for all $f, g \in C(X)$ by:

(2)
\begin{align} \quad \rho(f(x), g(x)) = \max_{x \in X} \{ \mid f(x) - g(x) \mid \} \end{align}

We verified that $\rho$ was indeed a metric and so $(C(X), \rho)$ is a metric space.

We will now define the concept of boundedness of a subset of $C(X)$.

 Definition: Let $(X, d)$ be a compact metric space and let $\Gamma \subseteq C(X)$. Then $\Gamma$ is said to be Bounded if there exists an $M \in \mathbb{R}$, $M > 0$ such that $\mid f(x) \mid \leq M$ for all $f \in \Gamma$ and for all $x \in X$.

For example, consider the metric space $([-2\pi, 2\pi], d)$ where $d$ is the usual Euclidean metric defined for all $x, y \in [-2\pi, 2\pi]$ by $d(x, y) = \mid x - y \mid$. Also consider the following subcollection:

(3)
\begin{align} \quad \Gamma = \{ f_r(x) = \cos (rx) : r \in \mathbb{R} \} \subset C(X) \end{align}

We claim that $\Gamma$ is bounded. To show this, take $M = 1 > 0$. Then since for all $a \in \mathbb{R}$ we have that $\mid \cos (a) \mid \leq 1$ we see that for all $f_r \in Gamma$ and for all $x \in X$ that then:

(4)
\begin{align} \quad \mid f_r(x) \mid = \mid \cos (rx) \mid \leq 1 = M \end{align}

Therefore $\Gamma$ is bounded.