User:Nuclear Catapult/sandbox

Linear Span
Given a vector space $V$ over a field $K$, the span of a set $S$ of vectors (not necessarily infinite) is defined to be the intersection $W$ of all subspaces of $V$ that contain $S$. $W$ is referred to as the subspace spanned by $S$, or by the vectors in $S$. Conversely, $S$ is called a spanning set of $W$, and we say that $S$ spans $W$.

Alternatively, the span of $S$ may be defined as the set of all finite linear combinations of elements (vectors) of $S$, which follows from the above definition.

$$ \operatorname{span}(S) = \left \{ {\left.\sum_{i=1}^k \lambda_i \mathbf v_i \;\right|\; k \in \N, \mathbf v_i  \in S, \lambda _i  \in K} \right \}.$$

In the case of infinite $S$, infinite linear combinations (i.e. where a combination may involve an infinite sum, assuming that such sums are defined somehow as in, say, a Banach space) are excluded by the definition; a generalization that allows these is not equivalent.