Closed linear span

In functional analysis, a branch of mathematics, the closed linear span of a set of vectors is the minimal closed set which contains the linear span of that set.

Definition

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

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

Notes

The linear span of a set is dense in the closed linear span. Moreover, as stated in the below lemma, the closed linear span is indeed the closure of the linear span.

Closed linear spans are important when dealing with closed linear subspaces (which are themselves highly important, consider Riesz's lemma).

A useful lemma

Let X be a normed space and let E be any non-empty subset of X. Then

(a) $\overline{\operatorname{Sp}}(E)$ is a closed linear subspace of X which contains E,

(b) $\overline{\operatorname{Sp}}(E)=\overline{\operatorname{Sp}(E)}$, viz. $\overline{\operatorname{Sp}}(E)$ is the closure of $\operatorname{Sp}(E)$,

(c) $E^\perp=(\operatorname{Sp}(E))^\perp=(\overline{\operatorname{Sp}(E)})^\perp.$

(So the usual way to find the closed linear span is to find the linear span first, and then the closure of that linear span.)

References

• Rynne & Youngson (2001). Linear functional analysis, Springer.

See also

• Linear span