Let $X$ be a metrizable topological space, and $C(X)$ the space of continuous functions. Is there a continuous norm (as function to $\mathbb{R}$) on $C(X)$? The topology is given by the family of seminorms (locally convex): $$ ||f||_K = \sup_{x \in K} \left| f(x) \right| $$ where $K \in X$ is compact.
I know that if the space $X$ is compact, then we have the norm. But how to prove that it is not in other case?
Let $X=\mathbb N$ with the usual metric. Then $C(X)$ can be identified with $\mathbb R ^{\mathbb N}$ and the topology you are considering is the topology of convergence at each point, i.e. the product topology. It is well known that there is no norm for this topology.