Given $E$, a locally convex space (l.c.s.) and $E\subseteq F$ where $F$ is another subspace of a larger vector space. The inclusion is strict since I know there exists a $y\in F\backslash E$.
I have two questions
Q1. Is it a known result that F also has a base of neighborhoods of 0 that are convex (in F) thereby making F also an l.c.s.?
My initial thoughts are that since every open set in $E$ (via the subspace topology) is $V \cap E$, where V is open in F, $V$ itself need not be convex. Yet, is it possible to "extend" the open sets in $E$ and "complete" them in some sense so that they are convex thereby forming a local convex base in F? Is this a known result? If it is, I would appreciate any reference (I have access to Munkres, Rudin's Functional Analysis) where it is proven.
Q2. Is $\overline E = E$? I know that there is a result that states that in a topological vector space, a hyperplane is either closed or dense. But the issue I am having here is that $F$ itself may be a proper subspace of a larger vector space. If $\overline E = E$, then I know for sure that $y\notin \overline E$ and hence I can separate $y$ and E by a closed hyperplane if $F$ is also a l.c.s.