Terminology: By a neighborhood of a point $x$ on a topological space, I mean any subset $V$ which contains an open set containing $x$. A set $B$ in a vector space $X$ is called balanced if $\lambda B \subseteq B$ for every $|\lambda| \le 1$.
Let $X$ be locally convex and suppose it is first countable. In particular, $X$ has a countable neighborhood basis of the origin. Because $X$ is locally convex, we know that (a) every element of such basis is balanced and (b) $X$ has a neighborhood basis of the origin consisting of balanced and convex sets.
Question: How can we find a countable neighborhood basis of the origin consisting of balanced and convex sets?
Best Answer
Let $(U_n)_n$ be any countable local base at $0$ for $X$.
As you stated under (b), for each $n$ we can find a convex balanced neighbourhood $V_n$ of $0$ such that $V_n \subseteq U_n$, as $X$ is locally convex.
It’s immediate that $(V_n)_n$ is your required local countable base at $0$.