Proposition 10

Every metrizable Montel space satisfies the first axiom of countability.

Let E be a metrizable Montel space. We know (II, p. 5) that E can be identified with a subspace of the product $F = \prod_{n \in \mathbf{N}} F_n$ of a sequence of normed spaces, and we can even assume that we have $\mathrm{pr}_n(E) = F_n$ for all $n \in \mathbf{N}$. If each of the metrizable spaces $F_n$ satisfies the first axiom of countability, then so does F (GT, IX, § 2, No. 8), hence also E.

We argue by reductio ad absurdum. Assume for example that $F_0$ does not satisfy the first axiom of countability. Let $B_0$ be the unit ball (closed) in $F_0$; this is a metric space which does not satisfy the first axiom of countability. We shall use the following lemma :