Span (mathematics)

In algebra, the span of a set of elements of a module or vector space is the set of all finite linear combinations of that set: it may equivalently be defined as the intersection of all submodules or subspaces containing the given set.

For S a subset of an R-module M we have


 * $$\langle S \rangle = \left\lbrace \sum_{i=1}^n r_i s_i : r_i \in R,~ s_i \in S \right\rbrace = \bigcap_{S \subseteq N; N \le M;} N .\,$$

If S is itself a submodule then $$S = \langle S \rangle$$.

The equivalence of the two definitions follows from the property of the submodules forming a closure system for which $$\langle \cdot \rangle$$ is the corresponding closure operator.