Compact Function Space

Last updated Nov 6, 2022

Let $X$ be a Topological Space and let $K \subset X$ be Compact. Let $(Y, ||\cdot||)$ be a Normed Vector Space. Then $C(K, Y)$ equipped with the Supremum Norm $||\cdot||_{\infty}$ is a Normed Vector Space. It is known as the Compact Function Space from $K$ to $Y$.

Note that Continuous Functions form a Vector Space so $C(K, Y)$ is indeed a Vector Space. Furthermore, by the Extreme Value Theorem, $||f(K)|| \subset \overline{\mathbb{R}{\geq 0}}$ is a Tightly Bounded Set for all $f \in C(K,Y)$. Therefore $\sup\limits ||f(K)|| =: ||f||{\infty}$ exists and is finite. Thus, $C(K,Y) = C_{b}(K, Y)$. Since the Bounded Continuous Function Space is a Normed Vector Space, the Compact Function Space is also one. $\blacksquare$

• Norms are Continuous
• Composition of Continuous Functions is Continuous