Let $E$ be a vector space over the filed of complex numbers. Suppose that $E$ is the union of an increasing sequence ($E_n \subset E_{n+1}$) of subspaces $E_n$, $n \in \mathbb{N}$ and that on each $E_n$ there is a structure of Fréchet space such that the natural injection of $E_n$ in $$E_{n+1}$ is an isomorphism.
Let $$\mathscr{B}=\{V\subset E: V \hbox{ is convex }, V \cap E_n \in \mathscr{F}_{E_n}(0) \hbox{ for all } n \in \mathbb{N}\},$$ where $\mathscr{F}_{E_n}(0)$ denotes the filter of neighborhooods of the origin in $E_n$.
My question: Let $V$ be a element of the family $\mathscr{B}$. Is there a balanced set $U$ belonging to the family $\mathscr{B}$ contained in $V$? Or are there spaces that do not fulfill this condition, in this case, what would be a counterexample?
The only thing I could think of was: Let $V \in \mathscr{B}$, since $V\cap E_n \in \mathscr{F}_{E_n}(0)$, for all $n \in \mathbb{N}$ there exists a balanced set (which we can assume convex) such that $U_n \in \mathscr{F}_{E_n}(0)$ such that $U_n \subset V\cap E_n\subset V$, but I don't know how to build a set $U$ satisfying the desired properties from tha family $\{U_n\}$.