Editing Linear span (section)

== Closed linear span (functional analysis) ==
In [[functional analysis]], a closed linear span of a [[Set (mathematics)|set]] of [[vector space|vectors]] is the minimal closed set which contains the linear span of that set.

Suppose that {{mvar|X}} is a normed vector space and let {{mvar|E}} be any non-empty subset of {{mvar|X}}. The '''closed linear span''' of {{mvar|E}}, denoted by <math>\overline{\operatorname{Sp}}(E)</math> or <math>\overline{\operatorname{Span}}(E)</math>, is the intersection of all the closed linear subspaces of {{mvar|X}} which contain {{mvar|E}}.

One mathematical formulation of this is

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

The closed linear span of the set of functions ''x<sup>n</sup>'' on the interval [0, 1], where ''n'' is a non-negative integer, depends on the norm used. If the [[Lp space#Lp spaces and Lebesgue integrals|''L''<sup>2</sup> norm]] is used, then the closed linear span is the [[Hilbert space]] of [[square-integrable function]]s on the interval. But if the [[maximum norm]] is used, the closed linear span will be the space of continuous functions on the interval. In either case, the closed linear span contains functions that are not polynomials, and so are not in the linear span itself. However, the [[cardinality]] of the set of functions in the closed linear span is the [[cardinality of the continuum]], which is the same cardinality as for the set of polynomials.

=== Notes ===
The linear span of a set is dense in the closed linear span. Moreover, as stated in the lemma below, the closed linear span is indeed the [[closure (mathematics)|closure]] of the linear span.

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

=== A useful lemma ===
Let {{mvar|X}} be a normed space and let {{mvar|E}} be any non-empty subset of {{mvar|X}}. Then
{{ordered list
 |list-style-type=lower-alpha
 | <math>\overline{\operatorname{Sp}}(E)</math> is a closed linear subspace of ''X'' which contains ''E'',
 | <math>\overline{\operatorname{Sp}}(E) = \overline{\operatorname{Sp}(E)}</math>, viz. <math>\overline{\operatorname{Sp}}(E)</math> is the closure of <math>\operatorname{Sp}(E)</math>,
 | <math>E^\perp = (\operatorname{Sp}(E))^\perp = \left(\overline{\operatorname{Sp}(E)}\right)^\perp.</math>
 | <math>(E^\perp)^\perp = ((\operatorname{Sp}(E))^\perp)^\perp = \overline{\operatorname{Sp}(E)}.</math>
}}

(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.)