Editing Topological vector space (section)

===Completeness and uniform structure===
{{Main|Complete topological vector space}}

The '''[[Complete topological vector space|canonical uniformity]]'''{{sfn|Schaefer|Wolff|1999|pp=12-19}} on a TVS <math>(X, \tau)</math> is the unique translation-invariant [[Uniform space|uniformity]] that induces the topology <math>\tau</math> on <math>X.</math>

Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into [[uniform space]]s. This allows one to talk{{clarify|date=September 2020}} about related notions such as [[Complete topological vector space|completeness]], [[uniform convergence]], Cauchy nets, and [[uniform continuity]], etc., which are always assumed to be with respect to this uniformity (unless indicated other). This implies that every Hausdorff topological vector space is [[Tychonoff space|Tychonoff]].{{sfn|Schaefer|Wolff|1999|p=16}} A subset of a TVS is [[Compact space|compact]] if and only if it is complete and [[totally bounded]] (for Hausdorff TVSs, a set being totally bounded is equivalent to it being [[Totally bounded space#In topological groups|precompact]]). But if the TVS is not Hausdorff then there exist compact subsets that are not closed. However, the closure of a compact subset of a non-Hausdorff TVS is again compact (so compact subsets are [[relatively compact]]).

With respect to this uniformity, a [[Net (mathematics)|net]] (or [[Sequence (mathematics)|sequence]]) <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> is '''Cauchy''' if and only if for every neighborhood <math>V</math> of <math>0,</math> there exists some index <math>n</math> such that <math>x_i - x_j \in V</math> whenever <math>i \geq n</math> and <math>j \geq n.</math>

Every [[Cauchy sequence]] is bounded, although Cauchy nets and Cauchy filters may not be bounded. A topological vector space where every Cauchy sequence converges is called '''[[sequentially complete]]'''; in general, it may not be complete (in the sense that all Cauchy filters converge).

The vector space operation of addition is uniformly continuous and an [[Open and closed map|open map]]. Scalar multiplication is [[Cauchy continuous]] but in general, it is almost never uniformly continuous. Because of this, every topological vector space can be completed and is thus a [[Dense set|dense]] [[linear subspace]] of a [[complete topological vector space]].

* Every TVS has a [[Complete topological vector space|completion]] and every Hausdorff TVS has a Hausdorff completion.{{sfn|Narici|Beckenstein|2011|pp=67-113}} Every TVS (even those that are Hausdorff and/or complete) has infinitely many non-isomorphic non-Hausdorff completions.
* A compact subset of a TVS (not necessarily Hausdorff) is complete.{{sfn|Narici|Beckenstein|2011|pp=115-154}} A complete subset of a Hausdorff TVS is closed.{{sfn|Narici|Beckenstein|2011|pp=115-154}}
* If <math>C</math> is a complete subset of a TVS then any subset of <math>C</math> that is closed in <math>C</math> is complete.{{sfn|Narici|Beckenstein|2011|pp=115-154}}
* A Cauchy sequence in a Hausdorff TVS <math>X</math> is not necessarily [[relatively compact]] (that is, its closure in <math>X</math> is not necessarily compact).
* If a Cauchy filter in a TVS has an [[Filters in topology|accumulation point]] <math>x</math> then it converges to <math>x.</math>
* If a series <math display=inline>\sum_{i=1}^{\infty} x_i</math> converges<ref group="note">A series <math display=inline>\sum_{i=1}^{\infty} x_i</math> is said to '''converge''' in a TVS <math>X</math> if the sequence of partial sums converges.</ref> in a TVS <math>X</math> then <math>x_{\bull} \to 0</math> in <math>X.</math>{{sfn|Swartz|1992|pp=27-29}}