Existence of continuous norm in topological vector spaces

Question

I am being asked to check whether at least one continuous norm exists in some locally convex Hausdorff topological vector spaces. I am not sure what is an efficient way (or useful theorem) that would help me prove/disprove it on each case.

Of course, if I prove that the space is normable, then such a norm exists (the one inducing the very topology of the space). However I imagine that there might be some spaces such that the topology admits other continuous norms that do not neccessarily induce the topology of the space. After all they are just functionals.

For instance, a space that I have to check whether this claim is true is $\mathbb{K}^X$ (the space of complex valued functions with domain $X$, where $X$ is a set) with the topology being that generated by the family of seminorms $\{p_{x}\}$ where $p_{x}(f)= |f(x)|$, $x \in X$. I am pretty sure that this space is not normable (using the theorem that if the topology of a Hausdorf l.c.s. can not be generated by a finite subset of seminorms then it is not normable).

Could you help me verify the existence of a continuous norm in this space? Or maybe giving me useful criteriorns to check this? Thanks.

Answer 1

For infinite $X$ the space $\mathbb C^X$ of all functions from $X$ to $\mathbb C$ endowed with the pointwise topology does not have continuous norms: The pointwise topology is generated by the seminorms $p_E(f)=\max\{|f(x)|: x\in E\}$ with $E\subseteq X$ finite. If the $q$ is any continuous seminorm, there are $E\subseteq X$ finite and $c>0$ with $q\le cp_E$ (this follows from the continuity of $q$ at $0$). For $x\in X\setminus E$ and $f=\delta_x$ defined by $\delta_x(x)=1$ and $\delta_x(y)=0$ for all $y\neq x$, you then have $0\le q(f)\le cp_E(f)=0$ so that $q(f)=0$ and $f\neq 0$. This shows that $q$ is not a proper norm.

Other examples of locally convex spaces without continuous norm appearing in functional analysis are:

• infinite dimensional Banach spaces with their weak topology
• spaces of continuous functions on infinite compact sets with the topology of pointwise convergence
• the space $\mathscr D'(\Omega)$ of distributions on an open subset $\Omega$ of $\mathbb R^n$ with the strong topology of uniform convergence on all bounded subsets of the space $\mathscr D(\Omega)$ of test functions.