I want to understand the definition of the $\varepsilon$-tensor product of two locally convex vector spaces. (Mainly as a hobby.)

Let $X,Y$ be locally convex vector spaces and let $B(X^{\ast},Y^{\ast})$ denote the space of separately continuous bilinear forms on $X^{\ast} \times Y^{\ast}$, where $X^{\ast}$, and $Y^{\ast}$ carry the weak$\ast$ topology. Let $A \subset X^{\ast}$ and $B \subset Y^{\ast}$ be two equicontinuos sets of functionals and let $\beta: X^{\ast} \times Y^{\ast} \rightarrow \mathbb{C}$ be an element of $B(X^{\ast}, Y^{\ast})$. Why is

$$ \sup_{(a,b) \in A \times B}{|\beta(a,b)|} < \infty\, ? $$

Here are my thoughts on this: I understand that, by using the Alaoglu-Bourbaki Theorem on polars and the assumption of equi-continuity, we can replace $A$ and $B$ by polars of neighbourhoods of zero in $X$ and $Y$ respectevely to obtain weak* compact, convex subsets of $X^{\ast}$ and $Y^{\ast}$ respectively. We may write the compact Hausdroff space $A$ as the union $$ A = \bigcup_{n \geq 1}{A_n}, \qquad A_n :=\{a \in A \,:\, \sup_{b \in B}{|\beta(a,b)|} \leq n\}.$$

Using that (locally) comapct Hausdorff spaces are Baire spaces, we can fix $n \geq 1$ and $x_0 \in A_n$ and an open zero-neighborhood $U \subset X^{\ast}$ such that $x_0 + U \subset A_n$. Now choose a finite subcover $A \subset \cup_{i=1}^n{(a_i + U)}$ of $ \cup_{a \in A}{(a + U)}$ and define $$ C:= \max_{1 \leq i \leq n}{\sup_{b \in B}{|\beta(a_i -x_0, b)|}}. $$ Then for arbitrary $b \in B$, $a \in A$, we write $a = a_i + x$ where $x \in U$, $1 \leq i \leq n$ and bound $$ |\beta(a,b)| =|\beta(a_i+x, b)| \leq |\beta(a_i-x_0, b)|+|\beta(x_0 + x, b)| \leq C+n. $$

Is this a valid argument?

The notes I am reading mention that one can use a "Banach-Steinhaus Theorem for compact convex sets", but my argument above doesn't use the convexity of $A$ or $B$. Which Theorem could the author of these notes (J.L. Taylor) be referring to?


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