Editing Banach space (section)

====Compact and convex subsets====

There is a compact subset <math>S</math> of <math>\ell^2(\N)</math> whose [[convex hull]] <math>\operatorname{co}(S)</math> is {{em|not}} closed and thus also {{em|not}} compact.<ref group=note name=ExampleCompactButHullIsNotCompact>Let <math>H</math> be the separable [[Hilbert space]] [[ℓ2 space|<math>\ell^2(\N)</math>]] of square-summable sequences with the usual norm <math>\|{\cdot}\|_2,</math> and let <math>e_n = (0, \ldots, 0, 1, 0, \ldots, 0)</math> be the standard [[orthonormal basis]] (that is, each <math>e_n</math> has zeros in every position except for a <math>1</math> in the <math>n</math><sup>th</sup>-position). The closed set <math>S = \{0\} \cup \{\tfrac{1}{n} e_n \mid n = 1, 2, \ldots\}</math> is compact (because it is [[Sequentially compact space|sequentially compact]]) but its convex hull <math>\operatorname{co} S</math> is {{em|not}} a closed set because the point <math display=inline>h := \sum_{n=1}^{\infty} \tfrac{1}{2^n} \tfrac{1}{n} e_n</math> belongs to the closure of <math>\operatorname{co} S</math> in <math>H</math> but <math>h \not\in\operatorname{co} S</math> (since every point <math>z=(z_1,z_2,\ldots) \in \operatorname{co} S</math> is a finite [[convex combination]] of elements of <math>S</math> and so <math>z_n = 0</math> for all but finitely many coordinates, which is not true of <math>h</math>). However, like in all [[Complete topological vector space|complete]] Hausdorff locally convex spaces, the {{em|closed}} convex hull <math>K := \overline{\operatorname{co}} S</math> of this compact subset is compact. The vector subspace <math>X := \operatorname{span} S = \operatorname{span} \{e_1, e_2, \ldots\}</math> is a [[pre-Hilbert space]] when endowed with the substructure that the Hilbert space <math>H</math> induces on it, but <math>X</math> is not complete and <math>h \not\in C := K \cap X</math> (since <math>h \not\in X</math>). The closed convex hull of <math>S</math> in <math>X</math> (here, "closed" means with respect to <math>X,</math> and not to <math>H</math> as before) is equal to <math>K \cap X,</math> which is not compact (because it is not a complete subset). This shows that in a Hausdorff locally convex space that is not complete, the closed convex hull of a compact subset might {{em|fail}} to be compact (although it will be [[Totally bounded space|precompact/totally bounded]]).</ref>{{sfn|Aliprantis|Border|2006|p=185}} 
However, like in all Banach spaces, the [[Closed convex hull|{{em|closed}} convex hull]] <math>\overline{\operatorname{co}} S</math> of this (and every other) compact subset will be compact.{{sfn|Trèves|2006|p=145}} In a normed space that is not complete then it is in general {{em|not}} guaranteed that <math>\overline{\operatorname{co}} S</math> will be compact whenever <math>S</math> is; an example<ref group=note name=ExampleCompactButHullIsNotCompact /> can even be found in a (non-complete) [[pre-Hilbert space|pre-Hilbert]] vector subspace of <math>\ell^2(\N).</math>