I'm wondering whether the following proposition is true:
Let $X, Y, Z$ be Frechet spaces. Let $A: X \times Y \to Z$ be continuous bilinear operator. If $x \in X$ and $\{y_n\}_{n\ge 1} \subset Y$ is bounded, is $\{A(x,y_n)\}_{n \ge 1}$ bounded in $Z$?
The specific case I'm wondering about is when Frechet spaces are space of symbols, and $A$ is a multiplication operator. In this special case, I can prove above property. I'm curious whether the generalization stated above is still true.
Also, in case above property is true, is it still true if we replace $x$ by a bounded sequence in $X$?
Thank you.