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 \[ \|f\|_{\sup} = \sup_{x \in X} \|f(x)\| .\]
Answer
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. $\Box$
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