Span (Mathematics)

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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

${\displaystyle ⟨S⟩=\{{\displaystyle \sum _{i=1}^n}{r}_i{s}_i:r_i\in R,s_i\in S\}=\bigcap _{S\subseteq N;N\le M}N.}$

We say that S spans, or is a spanning set for ${\displaystyle ⟨S⟩}$.

A basis is a linearly independent spanning set.

If S is itself a submodule then ${\displaystyle S=⟨S⟩}$.

The equivalence of the two definitions follows from the property of the submodules forming a closure system for which ${\displaystyle ⟨\cdot ⟩}$ is the corresponding closure operator.