This question has been motivated by weak* completeness of distributions.

According to the answer in the above post, any barrelled locally convex topological vector space $E$ satisfies the uniform boundedness principle for its continuous dual space $E'$.

My question is

Does the uniform boundedness principle hold for jointly continuous multilinear functionals on $E$ as well?

I have looked for any existing result on this, but it seems a bit elusive for me.

Could anyone please provide any information? I will move it to ME if this is not research-level question.

This question has been motivated by weak* completeness of distributions.

According to the answer in the above post, any barrelled locally convex topological vector space $E$ satisfies the uniform boundedness principle for its continuous dual space $E'$.

My question is

Does the uniform boundedness principle hold for jointly continuous multilinear functionals on $E$ as well?

I have looked for any existing result on this, but it seems a bit elusive for me.

Could anyone please provide any information? I will move it to ME if this is not research-level question.

OK, I will be more specific in my question. Let $E$ be a Frechet space. Consider a sequence of jointly continuous bilinear functions $ \{T_m : E \times E \to \mathbb{C}\}$ such that the limit \begin{equation} \lim\limits_{m \to \infty} T_m(v,w) \end{equation} exists for all $v,w \in E$. If we denote the limit by $T(v,w)$, then it is obviously a bilinear functional on $E$. However, my question is

Is $T : E \times E \to \mathbb{C}$ jointly continuous as well, in analogy with the known result on $F$-spaces?

I hope this clarifies my issue a bit more.