nLab
DF space

Idea

A DF space is a type of locally convex topological vector space. The basic idea is that a DF space is morally the strong dual? of a Fréchet space. That is, it has all the nice structure that such a dual would have but without the bother of actually having to be a dual space.

Definition

Definition

A locally convex topological vector space is a DF space if it possesses a fundamental sequence? of bounded sets and if every strongly bounded countable union of equicontinuous subsets of the dual is again equicontinuous.

Properties

1. The strong dual? of a metrisable locally convex topological vector space is a DF space.
2. Every normable space is a DF space.
3. Every infrabarrelled space that has a fundamental sequence? of bounded sets is a DF space.
4. In particular, every LB space? is a DF space, since any bounded set in an LB space is contained in one of the factors.

Proposition

A locally convex topological vector space that is metrisable and is a DF space is normable.

Proof

Let $E$ be a metrisable DF space.

To prove the result, we shall use a proposition recorded in Schaefer (IV.6.7). That says that a convex, circled subset $V$ of $E$ is a neighbourhood of $0$ if (and only if) for every convex, circled bounded subset $B\subseteq E$, $B\cap V$ is a $0$-neighbourhood in $B$.

As $E$ is metrisable, it has a countable $0$-neighbourhood base, say $(V_n)$. As $E$ is a DF space, it has a countable fundamental family of bounded sets, say $(B_n)$. Let us assume, without loss of generality, that this family is increasing.

For each $k\in \mathbb{N}$, as $B_k$ is bounded, there is some ${\lambda }_k>0$ such that $B_k\subseteq {\lambda }_k{V}_k$. Since, by assumption, the $B_k$ are an increasing family, we have $B_k\subseteq {\lambda }_l{V}_l$ for all $l\ge k$. Let $B≔\bigcap {\lambda }_k{V}_k$. This is a bounded set since it is contained in ${\lambda }_k{V}_k$ for each $k$. Let $l\in \mathbb{N}$ and consider $B_l\cap B$. We can write this as

$$B_l\cap \left(\bigcap _{k=1}^{l-1}{\lambda }_k{V}_k\right)\cap B_l\cap \left(\bigcap _{k\ge l}{\lambda }_k{V}_k\right).$$B_l \cap \big(\bigcap_{k=1}^{l-1} \lambda_k V_k\big) \cap B_l \cap \big( \bigcap_{k \ge l} \lambda_k V_k \big).

Since $B_l\subseteq {\lambda }_k{V}_k$ for $k\ge l$, the last part is unnecessary and so we see that

$$B_l\cap B=B_l\cap \left(\bigcap _{k=1}^{l-1}{\lambda }_k{V}_k\right).$$B_l \cap B = B_l \cap \big(\bigcap_{k=1}^{l-1} \lambda_k V_k \big).

This is then a finite intersection of open sets in $B_l$ and so is open in $B_l$.

As this holds for any of the $B_l$s, it holds for any bounded set, whence the proposition from Schaefer applies to show that $B$ is a $0$-neighbourhood. Thus $E$ possesses a bounded $0$-neighbourhood, whence is normable.

References

See functional analysis bibliography.