# The countable product of Fréchet spaces is a Fréchet space

Let $$\{E_n \; ; \; n \in \mathbb{N}\}$$ be a family of Fréchet spaces. I want to prove that the product $$E:= \prod_{n=1}^{\infty} E_n$$ is a Fréchet space, that is, $$E$$ is metrizable (Hausdorff space and admits a countable basis of neighborhoods of $$0\in E$$), complete and locally convex (admits a basis of neighborhoods of $$0\in E$$ consisting of convex sets).

I already know that $$E$$ is Hausdorff, locally convex and complete space. I don't know how to prove that $$E$$ is metrizable. How to proceed?

• How do you show the completeness without the metric, I wonder? – Henno Brandsma Jun 29 at 21:34
• Using the concepts of filters and Cauchy filters on topological vector spaces. For instante, see Exercise 5.5 for this book. – Guilherme de Loreno Jun 29 at 21:52

Let $$d_n$$ be one metric in each $$E_n$$. Then $$d(\tilde{x}, \tilde{y}):= \sum_{n=1}^{+\infty} \frac{d_n(x_n,y_n)}{1+d_n(x_n,y_n)}\frac{1}{2^n}$$ where $$\tilde{x}, \tilde{y}\in E$$, is a metric in $$E$$.
• And this metric define the topology of $E$ ? – Guilherme de Loreno Jun 29 at 18:41
• Isn't this in general the case? I mean if $X_n$ are metric spaces and $X$ is the countable product of them with the product topology then the function $$d(\tilde{x}, \tilde{y}):= \sum_{n=1}^{+\infty} \frac{d_n(x_n,y_n)}{1+d_n(x_n,y_n)}\frac{1}{2^n}$$ is a metric in $X$ and the induced topology is equivalent to the product topology. – alphaomega Jun 29 at 19:21
In this answer I deal with the metrisability dierctly, as a general result in metric spaces. Of course, if $$\mathcal{U}_n$$ is a countable local base of convex neighbourhoods of $$0$$ for $$E_n$$, (such that their intersection is $$\{0\}$$, which is equivalent to Hausdorffness in a TVS), the standard product local base for the $$(0)$$ product point
$$\{\prod_n U_n \mid \exists F \subseteq \Bbb N \text{ finite }: \forall n: ((n \in F) \to (U_n \in \mathcal{U_n})) \land (n \notin F ) \to (U_n = E_n)\}$$
consists of convex sets, is countable (as there are only countably many finite subsets of $$\Bbb N$$ and all $$\mathcal{U}_n$$ are countable), and intersects to $$\{(0)\}$$, so $$\prod_n E_n$$ is Fréchet.
The first proof I referred to has the added bonus that if all $$d_n$$ are complete, then so is the product sum-metric I define there. So that'll give completeness more easily, which I think is not yet apparent from the local base alone.