# Linear span

In the mathematical subfield of linear algebra or more generally functional analysis, the linear span (also called the linear hull) of a set of vectors in a vector space is the intersection of all subspaces containing that set. The linear span of a set of vectors is therefore a vector space.

## Definition

Given a vector space V over a field K, the span of a set S of vectors (not necessarily finite) 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 of S, which follows from the above definition.

$\operatorname {span} (S)=\left\{{\sum _{i=1}^{k}\lambda _{i}v_{i}{\Big |}k\in \mathbb {N} ,v_{i}\in S,\lambda _{i}\in \mathbf {K} }\right\}.$ In particular, if S is a finite subset of V, then the span of S is the set of all linear combinations of the elements of S. In the case of infinite S, infinite linear combinations (i.e. where a combination may involve an infinite sum, assuming such sums are defined somehow, e.g. if V is a Banach space) are excluded by the definition; a generalization that allows these is not equivalent.

## Examples

The real vector space R3 has {(2,0,0), (0,1,0), (0,0,1)} as a spanning set. This particular spanning set is also a basis. If (2,0,0) were replaced by (1,0,0), it would also form the canonical basis of R3.

Another spanning set for the same space is given by {(1,2,3), (0,1,2), (−1,1/2,3), (1,1,1)}, but this set is not a basis, because it is linearly dependent.

The set {(1,0,0), (0,1,0), (1,1,0)} is not a spanning set of R3; instead its span is the space of all vectors in R3 whose last component is zero.

## Theorems

Theorem 1: The subspace spanned by a non-empty subset S of a vector space V is the set of all linear combinations of vectors in S.

This theorem is so well known that at times it is referred to as the definition of span of a set.

Theorem 2: Every spanning set S of a vector space V must contain at least as many elements as any linearly independent set of vectors from V.

Theorem 3: Let V be a finite-dimensional vector space. Any set of vectors that spans V can be reduced to a basis for V by discarding vectors if necessary (i.e. if there are linearly dependent vectors in the set). If the axiom of choice holds, this is true without the assumption that V has finite dimension.

This also indicates that a basis is a minimal spanning set when V is finite-dimensional.

## Closed linear span

In functional analysis, a closed linear span of a set of vectors is the minimal closed set which contains the linear span of that set. Suppose that X is a normed vector space and let E be any non-empty subset of X. The closed linear span of E, denoted by ${\overline {\operatorname {Sp} }}(E)$ or ${\overline {\operatorname {Span} }}(E)$ , is the intersection of all the closed linear subspaces of X which contain E.

One mathematical formulation of this is

${\overline {\operatorname {Sp} }}(E)=\{u\in X|\forall \epsilon >0\,\exists x\in \operatorname {Sp} (E):\|x-u\|<\epsilon \}.$ 