Normable LF space?
Andrew Henderson 
Let $E$ be a locally convex Hausdorff space and consider its topological dual $E'_b$ endowed with the strong dual topology, i.e. uniform convergence on all bounded sets of $E$. In Trèves, "Topological Vector Spaces", p. 201 I found the following:
Exercise 19.4: $E'_b$ is metrizable $\Leftrightarrow$ $E'_b$ is normable $\Leftrightarrow$ $E$ is normable.
Exercise 19.7: Let $E = \bigcup_{k=1}^\infty E_k$ be an LF-space (inductive limit of Fréchet spaces $E_k$). Then $E'_b$ is Fréchet $\Leftrightarrow$ all the $E_k$ are normable.
In particular, consider $E := C_c(\mathbb{R})$ endowed with the inductive limit topology of $E_k := C_c(\mathbb{R}, [-k,k])$, the set of continuous functions with support in $[-k,k]$ endowed with the supremum norm. Thus, by Exercise 19.7: $E'_b$ is a Fréchet space and in particular metrizable. By Exercise 19.4: $E$ is normable.
How can it be that $C_c(\mathbb{R})$ with the inductive limit topology is normable? I thought the standard norm on $C_c(\mathbb{R})$ is the sup norm, but this induces another topology. Am I mixing things incorrectly together?
$\endgroup$1 Answer
$\begingroup$The assertions in exercise 19.4 are wrong.
Let $F$ be an infinite-dimensional normed space, and let $E$ be that space endowed with its weak topology. Then $E$ is a Hausdorff locally convex topological vector space that is not normable (every neighbourhood of $0$ in $E$ contains an infinite-dimensional linear subspace), but its strong dual $E_b'$ is just the dual of the normed space with its norm topology, hence normable, and a fortiori metrizable. The first equivalence "$E_b'$ is metrizable $\iff E_b'$ is normable" is also wrong, the class of DF-spaces is a class of HLCS whose strong dual is metrisable (namely their strong duals are Fréchet spaces), but only few of them have normable duals.
To have concrete examples, note that Montel spaces are reflexive, so every dual of an infinite-dimensional Fréchet-Montel space is a Hausdorff locally convex topological vector space whose strong dual is metrizable (since it's a Fréchet space) but not normable (since the only normable Montel spaces are the finite-dimensional spaces). Familiar examples of infinite-dimensional Fréchet-Montel spaces are
- the space $\mathscr{O}(U)$ of holomorphic functions on a non-empty open subset $U\subset \mathbb{C}^n$ in the topology of locally uniform convergence,
- the space $C^\infty(U)$ of infinitely differentiable functions on a non-empty open subset $U\subset \mathbb{R}^n$ in the topology of locally uniform convergence of all derivatives, and
- the space $\mathscr{S}(\mathbb{R}^n)$ of rapidly decreasing functions on $\mathbb{R}^n$.
