Does the uniform boundedness principle holds for multilinear maps as well?

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.

Answer 1

$\newcommand{\om}{\omega}$Let me answer your specific question.

The proof is similar to that of the uniform boundedness principle for linear functionals, but here using the identity \begin{equation} \begin{aligned} 4T_m(s,t)&=T_m(x+s,y+t)-T_m(x+s,y-t) \\ &-T_m(x-s,y+t)+T_m(x-s,y-t) \end{aligned} \tag{10}\label{10} \end{equation} for all $s,t,x,y$ in $E$.

Indeed, for natural $n$ let \begin{equation} F_n:=\{(v,w)\in E\times E\colon\,\sup_m|T_m(v,w)|\le n\}. \end{equation} Because the $T_m$'s are continuous, the sets $F_n$ are closed. Also, the condition \begin{equation} \lim_m T_m(v,w)=T(v,w) \tag{20}\label{20} \end{equation} for all $v,w$ in $E$ implies that $\bigcup_n F_n=E$. So, by the Baire category theorem, for some natural $n$, some $(x,y)\in E\times E$, and some balanced neighborhood $U$ of $0$ in $E$ we have \begin{equation} F_n\supseteq (x+U)\times (y+U). \end{equation} So, by \eqref{10}, $|T_m(s,t)|\le n$ for all $m$ and all $(s,t)\in U\times U$, and hence, in view of \eqref{20}, $|T(s,t)|\le n$ for all $(s,t)\in U\times U$.

Thus, $T$ is bounded on a neighborhood of $(0,0)$ and hence continuous. $\quad\Box$

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The same kind of argument holds for $k$-linear forms for any natural $k$. Then identity \eqref{10} will have to be replaced by the more general identity \begin{equation} \begin{alignedat}{2} &2^k\, T_m(s_1,\dots,s_k) &&\\ &=\sum_{(\om_1,\dots,\om_k)\in\{-1,1\}^k} &&(-1)^{1(\om_1=-1)+\dots+1(\om_k=-1)} \\ &&&\times T_m(x_1+\om_1 s_1,\dots,x_k+\om_k s_k) \end{alignedat} \tag{10a}\label{10a} \end{equation} for all $s_1,\dots,s_k,x_1,\dots,x_k$ in $E$. In turn, identity \eqref{10a} can be checked by writing the $k$-fold sum as a $k$-fold iterated sum.

Answer 2

It follows easily from the standard case (for linear mappings) that the pointwise limit is separately continuous and a standard result states that separately continuous multi-linear mappings (say on Frechet spaces) are continuous.

Answer 3

First, I think this is the type of questions that MO is for.

Second, Iosif Pinelis has already given an accepted answer, but for the record I give the following. "Uniform boundedness principle" and "Banach−Steinhaus" are closely related matters but some authors make a certain difference between the precise meaning of these phrases. OP's first question refers explicitly to the first without precisely specifying the meaning. The second more specific question refers to what is very close to the concept of "Banach−Steinhaus" in the sense that is used e.g. in Jarchow's Locally Convex Spaces on page 220. Now, the most general result in the direction of the second question, that I can "see", is the following

Theorem. Let $F$ be any sequentially complete Hausdorff locally convex space, and let $E_1,\ldots\,E_k$ be any metrizable and barrelled locally convex spaces. Then the space $L$ of separately continuous multilinear maps $E_1\times\ldots\,E_k\to F$ is sequentially complete for the topology of pointwise convergence. Moreover, every vector of $L$ is in fact (jointly) continuous.

I don't now whether there is any published proof of this, but a proof can be obtained by suitably putting together certain parts in the discussion on pages 13−22 in H. H. Keller's Differential Calculus in Locally Convex Spaces, Springer LNM 417, 1974. Further note for Hausdorff locally convex spaces the implications: Fréchet $\Rightarrow$ Baire $\Rightarrow$ barrelled, and that finite (and even countable) products of Fréchet spaces and barrelled spaces are of the same type, but that at least I do not know whether the same holds for "Baire". So in the case where $F$ is the scalar field (of real or complex numbers) or even a Banach space, the above theorem is only a slight generalization of the result for which Iosif Pinelis has already given a quite complete proof.