Is the evaluation map of tempered distribution $S^\prime\times S\to \mathbb R$ jointly continuous?

Question

$S=S(\mathbb R^d)$ denotes the Schwartz space of rapidely decreasing functions with its usual Fréchet space topology and the space of tempered distribution $S'$ is its dual with a suitable topology (strong or weak star). The question is whether the bilinear map $S'\times S\to \mathbb R$, $(u,f)\mapsto u(f)$ is continuous (for the product topology on $S'\times S$).

Answer 1

For any Hausdorff locally convex convex space $X$ and dual $X'$ endowed with the strong topology of uniform convergence on all bounded sets, the continuity of the evaluation map $Ev: X'\times X\to \mathbb R$, $(u,x)\mapsto u(x)$ implies that $X$ is normed (of course, $S$ is not normed so that this answers your question).

Indeed, the continuity of $Ev$ at $(0,0)$ implies that there are a bounded set $B$ and a $0$-neighbourhood $U$, both closed and absolutely convex, such that $Ev(B^\circ\times U)\subseteq (-1,1)$ (where $B^\circ$ denotes the absolute polar of $B$). This implies that $U\subseteq B$: Assume that $x\in U$ does not belong to $B$. Then Hahn-Banach yields $u\in X'$ with $u\in B^\circ$ and $|u(x)|>1$ which means $(u,x)\in B^\circ\times U$ but $|Ev(u,x)|\ge 1$, a contradiction.

Thus, $U$ is a bounded absolutely convex $0$-neoghbourhood and its Minkowski gauge (or functional) is a norm generating the topology of $X$.

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The fact just proved was of course known by the founders of the locally convex theory who thus studied other continuity properties of bilinear maps which are between joint and separate continuity (like hypo-continuity).