Bounded Continuous Function Space

Last updated Nov 6, 2022

Let $X$ be a Topological Space and let $(Y, ||\cdot||)$ be a Normed Vector Space. Then $$C_{b}(X, Y) := {f: X \to Y : \sup\limits_{x \in X} ||f(x)|| < \infty, f \text{ is continuous}}$$ is a Normed Vector Space with the Supremum Norm $||\cdot||_{\infty}$. It is called the Bounded Continuous Function Space from $X$ to $Y$.

Note that $C_{b}(X, Y)$ is simply the restrcition of the Vector Space $C(X, Y)$ to those with finite Supremum Norm. Since Restricting an Extended Norm to elements of Finite Norm form a Normed Vector Space, we have that $C_{b}(X, Y)$ is a Normed Vector Space. $\blacksquare$

• Continuous Functions form a Vector Space
• A Normed Vector Space is a Topological Vector Space