The Span of a Finite Family is the Smallest Subspace Containing It

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 $n$, let $v\in V^{n}$ be an \reftext{def:finite-tuple-power-2026a}{$n$-tuple} in $V$ with components $v_{k}$, and let $\operatorname{span}(v)$ be its \reftext{def:span-finite-family-2026b}{span}. Then the following hold.

\textbf{1. (Subspace)} $\operatorname{span}(v)$ is a \reftext{def:linear-subspace-2026a}{linear subspace} of $V$, and $v_{j}\in\operatorname{span}(v)$ for every $j\in[n]$.

\textbf{2. (Spanning)} By claim 1 and claim 1 of \ref{lem:subspace-inner-product-space-2026b}, the set $\operatorname{span}(v)$ is a vector space over $K$ under the operations of $V$ restricted to it. Let $v^{\ast}\in\operatorname{span}(v)^{n}$ be the $n$-tuple in $\operatorname{span}(v)$ with the same components $v_{k}$. Then $v^{\ast}$ \reftext{def:spanning-finite-family-2026a}{spans} that vector space.

\textbf{3. (Smallest such subspace)} If $W$ is a linear subspace of $V$ with $v_{k}\in W$ for every $k\in[n]$, then $\operatorname{span}(v)\subseteq W$.