Example of a LCS with a countably compact barrel

Question

I am self-studying topological vector spaces and I wonder if there is an example of a sequentially complete LCS with a countably compact barrel. I am a complete beginner and really can't think of any. Thanks all!

Answer 1

Here is a quite large family of examples which contains many nontrivial spaces. It should be somewhat beginner-friendly as it relies mostly on knowledge of normed spaces rather than general locally convex spaces.

Let $E$ be a Banach space and let $E'_s$ be its weak dual, i.e. $E'_s$ is the locally convex space obtained by endowing the vector space of all continuous linear functionals on $E$ with the topology of pointwise convergence on $E$. The space $E'_s$ can be shown to be sequentially complete and to have a compact barrel.

Proof of sequential completeness

Let $(x'_n)_{n\in\mathbf N}$ be a Cauchy sequence in $E'_s$. For every $x\in E$, the scalar sequence $(\langle x,x'_n\rangle)_{n\in\mathbf N}$ is Cauchy and thus is convergent. The scalar function $\displaystyle x\mapsto\lim_{n\to\infty}\langle x,x'_n\rangle$ on $E$ is obviously linear and, as a consequence of the Banach-Steinhaus theorem, can be seen to be continuous on $E$, meaning it is an element of $E'_s$. This shows that $(x'_n)_{n\in\mathbf N}$ is convergent in $E'_s$.

Description of a compact barrel

Let $B=\{x'\in E'_s~|~\forall x\in E,~|\langle x,x'\rangle|\leq\|x\|\}$ (i.e. $B$ is the closed unit ball of the Banach dual of $E$). It is easy to see that $B$ is a barrel of $E'_s$, and the fact that $B$ is compact is a consequence of the Banach-Alaoglu theorem.