Union of Balanced Sets in Vector Space is Balanced
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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.
$\blacksquare$