Let us consider the space $X=C_b(\mathbb{R},\mathbb{R})$ endowed with the following topology: $f_n\rightarrow f$ iff $f_n\rightarrow f$ uniformly on compact subsets of $\mathbb{R}$. This space is a locally convex topological vector space which is metrizable. Unless I'm really mistaken it is not a Frechet space, since a Cauchy sequence in $X$ could potentially converge to an unbounded (but continuous) function.
A barrelled space is a space for which each barrel (i.e. each convex, balanced, absorbing and closed subset) is in fact a neighborhood of zero. A fully barrelled space is a barrelled space for which each closed subspace is also barrelled. Frechet spaces (and therefore also Banach spaces) are examples of fully barrelled spaces. However, barrelled spaces need not be complete, but I have not found anywhere if fully barrelled spaces are complete.
My question is whether the space $X$ with this topology is a fully barrelled space.
The space is not even barrelled: $T=\{f\in X: |f(x)|\le 1$ for all $x\in\mathbb R\}$ is absolutely convex, closed, and absorbing in $X$, hence a barrel. But $T$ is not a $0$-neighbourhood because every $0$-neighbourhood $U$ in $X$ contains a set of the form $U_n=\{f\in X: |f(x)|\le 1/n$ for all $|x|\le n\}$ for some $n\in\mathbb N$ and no such $U_n$ is contained in $T$.