Cartesian product of Fréchet

Question

Let $F_{1},...,F_{n}$ be Fréchet spaces. Is the cartesian product $F_{1}\times\cdots \times F_{n}$ again a Fréchet space? In particular, is the cartesian product $\mathcal{S}(\mathbb{R}^{d})\times\mathcal{S}(\mathbb{R}^{d})$ of Schwartz spaces (of rapid decrease functions) again a Fréchet space?

Answer 1

Yes. An equivalent definition of $V$ being Fréchet is the existence of a countable collection of seminorms $||\cdot||_n$, $n\in\mathbb{N}=\{0,1,\ldots\}$, on $V$ such that:

1. The topology of $V$ is the locally convex topology defined by this collection of seminorms.
2. The distance $$ d(x,y)=\sum_{n=0}^{\infty}\ 2^{-n}\ \min(1,||x-y||_n) $$ makes $V$ a complete metric space.

So if you have two Fréchet spaces $V,W$ with respective collections of seminorms as above, $||\cdot||_{V,n}$ and $||\cdot||_{W,n}$, just take the collection $$ ||(v,w)||_{m,n}=\max(||v||_{V,m},||w||_{W,n})\ . $$ This gives a countable collection indexed by $\mathbb{N}^2$ on $V\times W$ which shows it is also Fréchet.

For a quick review of the definition of seminorms, locally convex topologies etc. see my answer Doubt in understanding Space $D(\Omega)$