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
We say that S spans, or is a spanning set for .
A basis is a linearly independent spanning set.
If S is itself a submodule then .
The equivalence of the two definitions follows from the property of the submodules forming a closure system for which is the corresponding closure operator.