Proposition 3

*Let $K$ be a valued locally compact and non-discrete division ring and $E$ be a Hausdorff topological vector space (resp. a topological vector space) over $K$. For every compact (resp. precompact) set $H$ in $E$, the balanced envelope of $H$ is compact (resp. precompact).

If $B$ denotes the ball $|\xi| \leq 1$ in $K$, the balanced envelope of $H$ is $H_1$, the image of $B \times H$ under the continuous mapping $m : (\lambda, x) \mapsto \lambda x$. If $E$ is Hausdorff, if $B$ is compact and if $H$ is compact then so is $B \times H$ and therefore $H_1$. If $H$ is precompact the restriction of $m$ to $B \times H$ is uniformly continuous (I, p. 5, Remark 2) and as $B \times H$ is precompact, so also is its image under $m$ (GT, II, § 4.2, prop. 2).

Note that the balanced envelope of a closed set is not necessarily closed. For example, in $\mathbf{R}^2$, the balanced envelope of the hyperbola defined by the equation $xy = 1$ is not closed.

The union of a family of balanced sets in $E$ is balanced, which implies that for every set $M$ of $E$ there is a largest balanced subset $N$ of $M$ called the balanced core of $M$; also $N$ is not empty if and only if $0 \in M$. To say that $x \in N$ means that for all $\lambda \in K$ such that $|\lambda| \leq 1$, we have $\lambda x \in M$, or again (if $0 \in M$) that, for all $\mu \in K$ with |μ| ≥ 1, we have x ∈ μM. If 0 ∈ M, the balanced core N of M is therefore the intersection $\bigcap_{|\mu| \geq 1} \mu M$. This shows in particular that if M is closed, so also is N.