Lemma: Fréchet multilinear continuity

A separately continuous finite multilinear map from a finite product of Fréchet spaces to a locally convex space is jointly continuous.

Proof. It suffices to obtain a local bound for each target seminorm. Induct on the number of arguments. For a fixed first argument, the map of the remaining arguments is jointly continuous by induction.

Choose countable neighborhood bases in the remaining Fréchet spaces. The sets of first arguments for which the target seminorm is bounded by one on a specified product of those neighborhoods are closed, balanced, and convex. Their countable dilates cover the first space. The Baire theorem therefore makes one such set have interior; its balanced convex structure then gives a neighborhood of zero. This supplies the required bound on a product of neighborhoods. Multilinearity and homogeneity give joint continuity.