Let $K$ be a \reftext{def:field-c54-2026b}{field}, let $V$ be a \reftext{def:vector-space-2026a}{vector space over $K$}, let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $[n]$ be the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by $n$, and let $v:[n]\to V$ be a map with values $v_{k}$.

The family $v$ \textbf{spans} $V$ if for every $u\in V$ there is a map $c:[n]\to K$ with

the sum being the \reftext{def:finite-sum-vector-space-2026a}{finite sum in $V$}.