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.
Let 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 of is a neighbourhood of if (and only if) for every convex, circled bounded subset , is a -neighbourhood in .
As is metrisable, it has a countable -neighbourhood base, say . As is a DF space, it has a countable fundamental family of bounded sets, say . Let us assume, without loss of generality, that this family is increasing.
For each , as is bounded, there is some such that . Since, by assumption, the are an increasing family, we have for all . Let . This is a bounded set since it is contained in for each . Let and consider . We can write this as
Since for , the last part is unnecessary and so we see that
This is then a finite intersection of open sets in and so is open in .
As this holds for any of the s, it holds for any bounded set, whence the proposition from Schaefer applies to show that is a -neighbourhood. Thus possesses a bounded -neighbourhood, whence is normable.