Linear Combination of Balanced Sets is Balanced

Theorem

Let $\GF \in \set {\R, \C}$.

Let $X$ be a vector space over $\GF$.

Let $\family {E_\alpha}_{\alpha \mathop \in A}$ be an $A$-indexed family of balanced sets.

Let $\lambda_\alpha \in \GF$ for each $\alpha \mathop \in A$.

Then:

$\ds \sum_{\alpha \mathop \in A} \lambda_\alpha E_\alpha$ is balanced.

Proof

Let $s \in \C$ have $\cmod s \le 1$.

Then, we have:

\(\ds s \sum_{\alpha \mathop \in A} \lambda_\alpha E_\alpha\)

\(=\)

\(\ds \sum_{\alpha \mathop \in A} \lambda_\alpha \paren {s E_\alpha}\)

Dilation of Subset of Vector Space Distributes over Sum: General Case

\(\ds \)

\(\subseteq\)

\(\ds \sum_{\alpha \mathop \in A} \lambda_\alpha E_\alpha\)

Definition of Balanced Set

So:

$\ds \sum_{\alpha \mathop \in A} \lambda_\alpha E_\alpha$ is balanced.

$\blacksquare$