Proposition 2

In a topological vector space $E$ over a valued division ring $K$, the closure of a balanced set $M$, is a balanced set.

If $B$ is the set of $\xi \in K$ with $|\xi| \leq 1$; then $B$ is closed in $K$. But $B \times M$ is mapped into $M$ by the continuous mapping $(\lambda, x) \mapsto \lambda x$; and therefore $B \times \overline{M}$ is mapped into $\overline{M}$ (GT, I, § 2.1, th. 1) which proves that $\overline{M}$ is balanced.

When $M$ is an arbitrary set in the vector space $E$ over a valued division ring $K$, the set $M_1$ of the $\lambda x$ with $x \in M$ and $\lambda \in K$ such that $|\lambda| \leq 1$, is clearly the smallest balanced set containing $M$; $M_1$ is called the balanced envelope of $M$.