# The space of continuous functions is a normed vector space

Given topological spaces $X$ and $Y$, define $C(X,Y) = \{f : X \to Y | \; f {\rm \; is \; continuous} \} .$
Let $X$ be a compact topological space and $V$ a normed vector space.

Prove that $C(X,V)$ becomes a normed vector space if we define addition and scalar multiplication in $C(X,V)$ by $(f + g)(x) = f(x) + g(x)$ $(c f)(x) = c f(x)$ for $f,g \in C(X,V)$, $c \in C$ and $x \in X$, and define the define sup norm of $f \in C(X,V)$ by $\|f\|_{\sup} = \sup_{x \in X} \|f(x)\| .$

First note that since $X$ is compact, every $f \in C(X,V)$ is bounded, i.e. $\|f\|_{\sup}<\infty$. (i) It is easy to see that for a continuous function $f:X\rightarrow Y$ $\|f\|_{\sup} =0 \Leftrightarrow f \equiv 0.$ (ii) For $\alpha \in \R$ $\| \alpha f\|_{\sup}= \sup_{x\in X} \|\alpha f\|= |\alpha| \sup_{x \in X} \|f\|=|\alpha| \|f\|_{\sup}.$ (iii) Let $f,g \in C(X,V)$. Then $\|f+g\|_{\sup} = \sup_{x \in X} \|f(x)+g(x)\|\leq \sup_{x\in X}\|f(x)\|+\sup_{x\in X}\|g(x)\|=\|f\|_{\sup}+\|g\|_{\sup}.$ Thus $C(X,Y)$ is a normed vector space.  $\Box$