This question has been motivated by [weak* completeness of distributions](https://math.stackexchange.com/questions/4615177/is-a-weak-limit-of-a-sequence-of-tempered-distributions-indeed-a-tempered-distr).


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.