Does a barrel contain a neighborhood of $0$?

Question

Suppose $X$ is topological vector space which is of the second category in itself. Let $K$ be a closed, convex, absorbing subset (a barrel) of $X$. Prove that $K$ contains a neighborhood of $0$.

Answer 1

As $K$ is absorbing, $X=\bigcup_n nK$, so some $nK$ must have nonempty interior, i.e. $x+U\subseteq nK$ for some $x\in X$ and a $0$-neighbourhood $U$. In addition, $-x\in mK$ for some $m$, so for $p=\max(m,n)$ you have $$ x+U\subseteq pK \, \text{ and } \, -x\in pK. $$ Hence, using the convexity of $K$, $$ U\subseteq -x+pK \subseteq pK+pK \subseteq 2p K. $$ As $\frac{1}{2p} U$ is a neighbourhood of $0$ too, you are done.