Definition:Linear Combination of Subsets of Vector Space/General Case

Definition

Let $K$ be a field.

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

Let $I$ be an indexing set.

For each $\alpha \in I$, let $E_\alpha$ be a subset of $X$ and $\lambda_\alpha \in K$.

We define the linear combination $\ds \sum_{\alpha \mathop \in I} \lambda_\alpha E_\alpha$ by:

$\ds \sum_{\alpha \mathop \in I} \lambda_\alpha E_\alpha = \set {\sum_{i \in F} \lambda_i x_i : F \text { is a finite subset of } I, \, x_i \in E_i \text { for each } i \in I}$

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