Union of Balanced Sets in Vector Space is Balanced

Theorem

Let $\Bbb F \in \set {\R, \C}$.

Let $X$ be a vector space over $\Bbb F$.

Let $\sequence {E_\alpha}_{\alpha \mathop \in I}$ be an $I$-indexed family of balanced subsets of $X$.

Then:

$\ds E = \bigcup_{\alpha \mathop \in I} E_\alpha$ is balanced.

Proof

Let $x \in E$.

Let $\lambda \in \Bbb F$ have $\cmod \lambda \le 1$.

We aim to show that $\lambda x \in E$.

Since $x \in E$, there exists $\alpha \in I$ with $x \in E_\alpha$.

Since $E_\alpha$ is balanced, we have $\lambda x \in E_\alpha$.

So:

$\ds \lambda x \in \bigcup_{\alpha \mathop \in I} E_\alpha = E$

Since $x$ and $\lambda \in \Bbb F$ with $\cmod \lambda \le 1$ were arbitrary, $E$ is balanced.

$\blacksquare$