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

Theorem

Let $K$ be a field.

Let $X$ be a vector space over $K$.

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

Let $\lambda \in K$.

Then:

$\ds \lambda \sum_{\alpha \mathop \in A} E_\alpha = \sum_{\alpha \mathop \in A} \paren {\lambda E_\alpha}$

Proof

Let $x \in X$.

We have:

$\ds x \in \lambda \sum_{\alpha \mathop \in A} E_\alpha$

if and only if there exists:

a finite subset $F \subseteq E_\alpha$

$x_\alpha \in F$ for each $\alpha \in F$

such that:

$\ds x = \lambda \sum_{\alpha \in F} x_\alpha$

This is equivalent to:

$\ds x = \sum_{\alpha \in F} \lambda x_\alpha$

for a finite subset $F \subseteq E_\alpha$, with $x_\alpha \in F$ for each $\alpha \in F$.

Hence we obtain:

$\ds x \in \lambda \sum_{\alpha \mathop \in A} E_\alpha$ if and only if $x \in \sum_{\alpha \mathop \in A} \paren {\lambda E_\alpha}$

so that:

$\ds \lambda \sum_{\alpha \mathop \in A} E_\alpha = \sum_{\alpha \mathop \in A} \paren {\lambda E_\alpha}$

$\blacksquare$