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{{Short description|A TVS where points that get progressively closer to each other will always converge to a point}}
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In [[Functional analysis|functional analysis]] and related areas of [[Mathematics|mathematics]], a '''complete topological vector space''' is a [[Topological vector space|topological vector space]] (TVS) with the property that whenever points get progressively closer to each other, then there exists some point <math>x</math> towards which they all get closer. 
The notion of "points that get progressively closer" is made rigorous by {{em|Cauchy nets}} or {{em|Cauchy filters}}, which are generalizations of {{em|[[Cauchy sequence]]s}}, while "point <math>x</math> towards which they all get closer" means that this Cauchy net or filter converges to <math>x.</math> 
The notion of completeness for TVSs uses the theory of [[Uniform space|uniform space]]s as a framework to generalize the notion of [[Complete metric space|completeness for metric spaces]]. 
But unlike metric-completeness, TVS-completeness does not depend on any metric and is defined for {{em|all}} TVSs, including those that are not [[Metrizable space|metrizable]] or [[Hausdorff space|Hausdorff]].

Completeness is an extremely important property for a topological vector space to possess. 
The notions of completeness for normed spaces and [[Metrizable topological vector space|metrizable TVS]]s, which are commonly defined in terms of completeness of a particular norm or metric, can both be reduced down to this notion of TVS-completeness &ndash; a notion that is independent of any particular norm or metric. 
A [[Metrizable topological vector space|metrizable topological vector space]] <math>X</math> with a translation invariant metric<ref group=note name="translation invariant metric">A metric <math>D</math> on a vector space <math>X</math> is said to be '''translation invariant''' if <math>D(x, y) = D(x + z, y + z)</math> for all vectors <math>x, y, z \in X.</math> A metric that is induced by a norm is always translation invariant.</ref> <math>d</math> is complete as a TVS if and only if <math>(X, d)</math> is a [[Complete metric space|complete metric space]], which by definition means that every <math>d</math>-[[Cauchy sequence]] converges to some point in <math>X.</math> 
Prominent examples of complete TVSs that are also [[Metrizable topological vector space|metrizable]] include all [[F-space]]s and consequently also all [[Fréchet space]]s, [[Banach space]]s, and [[Hilbert space]]s. 
Prominent examples of complete TVS that are (typically) {{em|not}} metrizable include strict [[LF-space]]s such as the [[Spaces of test functions and distributions|space of test functions]] <math>C_c^\infty(U)</math> with it canonical LF-topology, the [[Strong dual space|strong dual space]] of any non-normable [[Fréchet space]], as well as many other [[Polar topology|polar topologies]] on continuous dual space or other [[Topologies on spaces of linear maps|topologies on spaces of linear maps]].

Explicitly, a topological vector spaces (TVS) is '''complete''' if every [[Net (mathematics)|net]], or equivalently, every [[Filters in topology|filter]], that is Cauchy with respect to the space's {{em|canonical uniformity}} necessarily converges to some point. Said differently, a TVS is complete if its canonical uniformity is a complete uniformity. 
The '''canonical uniformity''' on a TVS <math>(X, \tau)</math> is the unique<ref group=note>Completeness of normed spaces and metrizable TVSs are defined in terms of [[Norm (mathematics)|norms]] and [[Metric (mathematics)|metrics]]. In general, many different norms (for example, equivalent norms) and metrics may be used to determine completeness of such space. This stands in contrast to the uniqueness of this translation-invariant canonical uniformity.</ref> translation-invariant [[Uniform space|uniformity]] that induces on <math>X</math> the topology <math>\tau.</math> 
This notion of "TVS-completeness" depends {{em|only}} on vector subtraction and the topology of the TVS; consequently, it can be applied to all TVSs, including those whose topologies can not be defined in terms [[Metric (mathematics)|metrics]] or [[Pseudometric space|pseudometrics]]. 
A first-countable TVS is complete if and only if every Cauchy sequence (or equivalently, every elementary Cauchy filter) converges to some point.

Every topological vector space <math>X,</math> even if it is not [[Metrizable space|metrizable]] or not [[Hausdorff space|Hausdorff]], has a {{em|completion}}, which by definition is a complete TVS <math>C</math> into which <math>X</math> can be TVS-embedded as a [[Dense set|dense]] vector subspace. Moreover, every Hausdorff TVS has a {{em|Hausdorff}} completion, which is necessarily unique [[Up to|up to]] TVS-isomorphism. However, as discussed below, all TVSs have infinitely many non-Hausdorff completions that are {{em|not}} TVS-isomorphic to one another.

==Definitions==

{{Main|Net (mathematics)|Filters in topology}}

This section summarizes the definition of a complete [[Topological vector space|topological vector space]] (TVS) in terms of both [[Net (mathematics)|nets]] and prefilters. 
Information about convergence of nets and filters, such as definitions and properties, can be found in the article about [[Filters in topology|filters in topology]].

Every topological vector space (TVS) is a commutative [[Topological group|topological group]] with identity under addition and the canonical uniformity of a TVS is defined {{em|entirely}} in terms of subtraction (and thus addition); scalar multiplication is not involved and no additional structure is needed.

===Canonical uniformity===

The '''{{em|diagonal}}''' of <math>X</math> is the set{{sfn|Schaefer|Wolff|1999|pp=1-11}} 
<math display=block>\Delta_X ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{(x, x) : x \in X\}</math> 
and for any <math>N \subseteq X,</math> the '''{{em|{{visible anchor|canonical entourage}}}}'''/'''{{em|{{visible anchor|canonical vicinity|text=vicinity}} around <math>N</math>}}''' is the set
<math display=block>\begin{alignat}{4}
\Delta_X(N) 
~&~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{(x, y) \in X \times X ~:~ x - y \in N\} = \bigcup_{y \in X} [(y + N) \times \{y\}] \\
&= \Delta_X + (N \times \{0\})
\end{alignat}</math>
where if <math>0 \in N</math> then <math>\Delta_X(N)</math> contains the diagonal <math>\Delta_X(\{0\}) = \Delta_X.</math>

If <math>N</math> is a [[Symmetric set|symmetric set]] (that is, if <math>- N = N</math>), then <math>\Delta_X(N)</math> is '''{{em|symmetric}}''', which by definition means that <math>\Delta_X(N) = \left(\Delta_X(N)\right)^{\operatorname{op}}</math> holds where <math>\left(\Delta_X(N)\right)^{\operatorname{op}} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left\{(y, x) : (x, y) \in \Delta_X(N)\right\},</math> and in addition, this symmetric set's {{em|composition}} with itself is: 
<math display=block>\begin{alignat}{4}
\Delta_X(N) \circ \Delta_X(N) 
~&~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left\{(x, z) \in X \times X ~:~ \text{ there exists } y \in X \text{ such that } x, z \in y + N\right\} = \bigcup_{y \in X} [(y + N) \times (y + N)] \\
&= \Delta_X + (N \times N).
\end{alignat}</math>

If <math>\mathcal{L}</math> is any neighborhood basis at the origin in <math>(X, \tau)</math> then the [[Family of sets|family of subsets]] of <math>X \times X:</math> 
<math display=block>\mathcal{B}_{\mathcal{L}} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left\{\Delta_X(N) : N \in \mathcal{L}\right\}</math>
is a prefilter on <math>X \times X.</math> 
If <math>\mathcal{N}_{\tau}(0)</math> is the neighborhood filter at the origin in <math>(X, \tau)</math> then <math>\mathcal{B}_{\mathcal{N}_{\tau}(0)}</math> forms a base of entourages for a [[Uniform space|uniform structure]] on <math>X</math> that is considered canonical.{{sfn|Edwards|1995|p=61}} 
Explicitly, by definition, {{em|the '''{{visible anchor|canonical uniformity}}''' on <math>X</math> induced by}} <math>(X, \tau)</math>{{sfn|Edwards|1995|p=61}} is the [[Filter (set theory)|filter]] <math>\mathcal{U}_{\tau}</math> on <math>X \times X</math> [[Filters in topology|generated by the above prefilter]]: 
<math display=block>\mathcal{U}_{\tau} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \mathcal{B}_{\mathcal{N}_{\tau}(0)}^{\uparrow} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left\{S \subseteq X \times X ~:~ N \in \mathcal{N}_{\tau}(0) \text{ and } \Delta_X(N) \subseteq S\right\}</math>
where <math>\mathcal{B}_{\mathcal{N}_{\tau}(0)}^{\uparrow}</math> denotes the {{em|upward closure}} of <math>\mathcal{B}_{\mathcal{N}_{\tau}(0)}</math> in <math>X \times X.</math> 
The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin. If <math>\mathcal{L}</math> is any neighborhood basis at the origin in <math>(X, \tau)</math> then the filter on <math>X \times X</math> generated by the prefilter <math>\mathcal{B}_{\mathcal{L}}</math> is equal to the canonical uniformity <math>\mathcal{U}_{\tau}</math> induced by <math>(X, \tau).</math>

===Cauchy net===

The general theory of [[Uniform space|uniform space]]s has its own definition of a "Cauchy prefilter" and "Cauchy net". For the canonical uniformity on <math>X,</math> these definitions reduce down to those given below.

Suppose <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> is a net in <math>X</math> and <math>y_{\bull} = \left(y_j\right)_{j \in J}</math> is a net in <math>Y.</math> 
The product <math>I \times J</math> becomes a [[Directed set|directed set]] by declaring <math>(i, j) \leq \left(i_2, j_2\right)</math> if and only if <math>i \leq i_2</math> and <math>j \leq j_2.</math> 
Then 
<math display=block>x_{\bull} \times y_{\bull} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(x_i, y_j\right)_{(i, j) \in I \times J}</math>
denotes the ('''Cartesian''') '''{{visible anchor|product net}}''', where in particular <math display=inline>x_{\bull} \times x_{\bull} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(x_i, x_j\right)_{(i, j) \in I \times I}.</math> If <math>X = Y</math> then the image of this net under the vector addition map <math>X \times X \to X</math> denotes the '''{{visible anchor|sum|sum of nets}}''' of these two nets:{{sfn|Narici|Beckenstein|2011|pp=47-66}} 
<math display=block>x_{\bull} + y_{\bull} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(x_i + y_j\right)_{(i, j) \in I \times J}</math>
and similarly their '''{{visible anchor|difference|difference of nets}}''' is defined to be the image of the product net under the vector subtraction map <math>(x, y) \mapsto x - y</math>:
<math display=block>x_{\bull} - y_{\bull} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(x_i - y_j\right)_{(i, j) \in I \times J}.</math> 
In particular, the notation <math>x_{\bull} - x_{\bull} = \left(x_i\right)_{i \in I} - \left(x_i\right)_{i \in I}</math> denotes the <math>I^2</math>-indexed net <math>\left(x_i - x_j\right)_{(i, j) \in I \times I}</math> and not the <math>I</math>-indexed net <math>\left(x_i - x_i\right)_{i \in I} = (0)_{i \in I}</math> since using the latter as the definition would make the notation useless.

A [[Net (mathematics)|net]] <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> in a TVS <math>X</math> is called a '''Cauchy net'''{{sfn|Narici|Beckenstein|2011|p=48}} if 
<math display=block>x_{\bull} - x_{\bull} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(x_i - x_j\right)_{(i, j) \in I \times I} \to 0 \quad \text{ in } X.</math> 
Explicitly, this means that for every neighborhood <math>N</math> of <math>0</math> in <math>X,</math> there exists some index <math>i_0 \in I</math> such that <math>x_i - x_j \in N</math> for all indices <math>i, j \in I</math> that satisfy <math>i \geq i_0</math> and <math>j \geq i_0.</math> 
It suffices to check any of these defining conditions for any given neighborhood basis of <math>0</math> in <math>X.</math> 
A '''[[Cauchy sequence]]''' is a sequence that is also a Cauchy net.

If <math>x_{\bull} \to x</math> then <math>x_{\bull} \times x_{\bull} \to (x, x)</math> in <math>X \times X</math> and so the continuity of the vector subtraction map <math>S : X \times X \to X,</math> which is defined by <math>S(x, y) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ x - y,</math> guarantees that <math>S\left(x_{\bull} \times x_{\bull}\right) \to S(x, x)</math> in <math>X,</math> where <math>S\left(x_{\bull} \times x_{\bull}\right) = \left(x_i - x_j\right)_{(i, j) \in I \times I} = x_{\bull} - x_{\bull}</math> and <math>S(x, x) = x - x = 0.</math>
This proves that every convergent net is a Cauchy net. 
By definition, a space is called {{em|complete}} if the converse is also always true. 
That is, <math>X</math> is complete if and only if the following holds: 
:whenever <math>x_{\bull}</math> is a net in <math>X,</math> then <math>x_{\bull}</math> converges (to some point) in <math>X</math> if and only if <math>x_{\bull} - x_{\bull} \to 0</math> in <math>X.</math> 
A similar characterization of completeness holds if filters and prefilters are used instead of nets.

A series <math>\sum_{i=1}^{\infty} x_i</math> is called a '''{{visible anchor|Cauchy series}}''' (respectively, a '''{{visible anchor|convergent series}}''') if the sequence of partial sums <math>\left(\sum_{i=1}^n x_i\right)_{n=1}^{\infty}</math> is a [[Cauchy sequence]] (respectively, a convergent sequence).{{sfn|Zălinescu|2002|pp=1-23}} Every convergent series is necessarily a Cauchy series. In a complete TVS, every Cauchy series is necessarily a convergent series.

===Cauchy filter and Cauchy prefilter===
{{anchor|Cauchy filter|Cauchy prefilter}}

A prefilter <math>\mathcal{B}</math> on a [[Topological vector space|topological vector space]] <math>X</math> is called a '''Cauchy prefilter'''{{sfn|Narici|Beckenstein|2011|pp=48-51}} if it satisfies any of the following equivalent conditions: 
<ol>
<li><math>\mathcal{B} - \mathcal{B} \to 0</math> in <math>X.</math> 
* The family <math>\mathcal{B} - \mathcal{B} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{B - C : B, C \in \mathcal{B}\}</math> is a prefilter.
* Explicitly, <math>\mathcal{B} - \mathcal{B} \to 0</math> means that for every neighborhood <math>N</math> of the origin in <math>X,</math> there exist <math>B, C \in \mathcal{B}</math> such that <math>B - C \subseteq N.</math></li>
<li><math>\{B - B : B \in \mathcal{B}\} \to 0</math> in <math>X.</math> 
* The family <math>\{B - B : B \in \mathcal{B}\}</math> is a prefilter equivalent to <math>\mathcal{B} - \mathcal{B}</math> (''equivalence'' means these prefilters generate the same filter on <math>X</math>).
* Explicitly, <math>\{B - B : B \in \mathcal{B}\} \to 0</math> means that for every neighborhood <math>N</math> of the origin in <math>X,</math> there exists some <math>B \in \mathcal{B}</math> such that <math>B - B \subseteq N.</math></li>
<li>For every neighborhood <math>N</math> of the origin in <math>X,</math> <math>\mathcal{B}</math> contains some <math>N</math>-small set (that is, there exists some <math>B \in \mathcal{B}</math> such that <math>B - B \subseteq N</math>).{{sfn|Narici|Beckenstein|2011|pp=48–51}}
* A subset <math>B \subseteq X</math> is called '''<math>N</math>-small''' or '''{{visible anchor|small of order}} <math>N</math>'''{{sfn|Narici|Beckenstein|2011|pp=48-51}} if <math>B - B \subseteq N.</math></li>
<li>For every neighborhood <math>N</math> of the origin in <math>X,</math> there exists some <math>x \in X</math> and some <math>B \in \mathcal{B}</math> such that <math>B \subseteq x + N.</math>{{sfn|Narici|Beckenstein|2011|pp=48-51}}
* This statement remains true if "<math>B \subseteq x + N</math>" is replaced with "<math>x + B \subseteq N.</math>"</li>
<li>Every neighborhood of the origin in <math>X</math> contains some subset of the form <math>x + B</math> where <math>x \in X</math> and <math>B \in \mathcal{B}.</math></li>
</ol>
It suffices to check any of the above conditions for any given neighborhood basis of <math>0</math> in <math>X.</math> 
A '''Cauchy filter''' is a Cauchy prefilter that is also a [[Filter (set theory)|filter]] on <math>X.</math>

If <math>\mathcal{B}</math> is a prefilter on a topological vector space <math>X</math> and if <math>x \in X,</math> then <math>\mathcal{B} \to x</math> in <math>X</math> if and only if <math>x \in \operatorname{cl} \mathcal{B}</math> and <math>\mathcal{B}</math> is Cauchy.{{sfn|Narici|Beckenstein|2011|pp=47-66}}

===Complete subset===

For any <math>S \subseteq X,</math> a prefilter <math>\mathcal{C}</math> '''{{em|on <math>S</math>}}''' is necessarily a subset of <math>\wp(S)</math>; that is, <math>\mathcal{C} \subseteq \wp(S).</math>

A subset <math>S</math> of a TVS <math>(X, \tau)</math> is called a '''{{visible anchor|complete subset}}''' if it satisfies any of the following equivalent conditions: 
<ol>
<li>Every Cauchy prefilter <math>\mathcal{C} \subseteq \wp(S)</math> on <math>S</math> converges to at least one point of <math>S.</math>
* If <math>X</math> is Hausdorff then every prefilter on <math>S</math> will converge to at most one point of <math>X.</math> But if <math>X</math> is not Hausdorff then a prefilter may converge to multiple points in <math>X.</math> The same is true for nets.</li>
<li>Every Cauchy net in <math>S</math> converges to at least one point of <math>S.</math></li>
<li><math>S</math> is a complete uniform space (under the point-set topology definition of "complete uniform space") when <math>S</math> is endowed with the uniformity induced on it by the canonical uniformity of <math>X.</math></li>
</ol>

The subset <math>S</math> is called a '''{{visible anchor|sequentially complete subset}}''' if every Cauchy sequence in <math>S</math> (or equivalently, every elementary Cauchy filter/prefilter on <math>S</math>) converges to at least one point of <math>S.</math>

Importantly, {{em|convergence to points outside of <math>S</math> does not prevent a set from being complete}}: If <math>X</math> is not Hausdorff and if every Cauchy prefilter on <math>S</math> converges to some point of <math>S,</math> then <math>S</math> will be complete even if some or all Cauchy prefilters on <math>S</math> {{em|also}} converge to points(s) in <math>X \setminus S.</math> In short, there is no requirement that these Cauchy prefilters on <math>S</math> converge {{em|only}} to points in <math>S.</math> The same can be said of the convergence of Cauchy nets in <math>S.</math>

As a consequence, if a TVS <math>X</math> is {{em|not}} Hausdorff then every subset of the closure of <math>\{0\}</math> in <math>X</math> is complete because it is compact and every compact set is necessarily complete. In particular, if <math>\varnothing \neq S \subseteq \operatorname{cl}_X \{0\}</math> is a proper subset, such as <math>S = \{0\}</math> for example, then <math>S</math> would be complete even though {{em|every}} Cauchy net in <math>S</math> (and also every Cauchy prefilter on <math>S</math>) converges to {{em|every}} point in <math>\operatorname{cl}_X \{0\},</math> including those points in <math>\operatorname{cl}_X \{0\}</math> that do not belong to <math>S.</math> 
This example also shows that complete subsets (and indeed, even compact subsets) of a non-Hausdorff TVS may fail to be closed. For example, if <math>\varnothing \neq S \subseteq \operatorname{cl}_X \{0\}</math> then <math>S = \operatorname{cl}_X \{0\}</math> if and only if <math>S</math> is closed in <math>X.</math>

===Complete topological vector space===

A [[Topological vector space|topological vector space]] <math>X</math> is called a '''{{visible anchor|complete topological vector space}}''' if any of the following equivalent conditions are satisfied:

<ol>
<li><math>X</math> is a complete uniform space when it is endowed with its canonical uniformity.
* In the general theory of [[Uniform space|uniform space]]s, a uniform space is called a complete uniform space if each Cauchy filter on <math>X</math> converges to some point of <math>X</math>  in the topology induced by the uniformity. When <math>X</math> is a TVS, the topology induced by the canonical uniformity is equal to <math>X</math>'s given topology (so convergence in this induced topology is just the usual convergence in <math>X</math>).</li>
<li><math>X</math> is a complete subset of itself.</li>
<li>There exists a neighborhood of the origin in <math>X</math> that is also a complete subset of <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=48-51}}
* This implies that every [[Locally compact space|locally compact]] TVS is complete (even if the TVS is not Hausdorff).</li>
<li>Every Cauchy prefilter <math>\mathcal{C} \subseteq \wp(X)</math> on <math>X</math> converges in <math>X</math> to at least one point of <math>X.</math>
* If <math>X</math> is Hausdorff then every prefilter on <math>X</math> will converge to at most one point of <math>X.</math> But if <math>X</math> is not Hausdorff then a prefilter may converge to multiple points in <math>X.</math> The same is true for nets.</li>
<li>Every Cauchy [[Filter (set theory)|filter]] on <math>X</math> converges in <math>X</math> to at least one point of <math>X.</math></li>
<li>Every Cauchy net in <math>X</math> converges in <math>X</math> to at least one point of <math>X.</math></li>
</ol>

where if in addition <math>X</math> is [[Metrizable topological vector space|pseudometrizable]] or metrizable (for example, a normed space) then this list can be extended to include:

<ol start=7>
<li><math>X</math> is sequentially complete.</li>
</ol>

A topological vector space <math>X</math> is '''{{visible anchor|sequentially complete}}''' if any of the following equivalent conditions are satisfied:
<ol>
<li><math>X</math> is a sequentially complete subset of itself.</li>
<li>Every Cauchy sequence in <math>X</math> converges in <math>X</math> to at least one point of <math>X.</math></li>
<li>Every elementary Cauchy prefilter on <math>X</math> converges in <math>X</math> to at least one point of <math>X.</math></li>
<li>Every elementary Cauchy filter on <math>X</math> converges in <math>X</math> to at least one point of <math>X.</math></li>
</ol>

==Uniqueness of the canonical uniformity==

The existence of the canonical uniformity was demonstrated above by defining it. The theorem below establishes that the canonical uniformity of any TVS <math>(X, \tau)</math> is the only uniformity on <math>X</math> that is both (1) translation invariant, and (2) generates on <math>X</math> the topology <math>\tau.</math>

{{Math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|pp=12-19}}|note=Existence and uniqueness of the canonical uniformity|math_statement=
The topology of any TVS can be derived from a unique translation-invariant uniformity. If <math>\mathcal{N}(0)</math> is any neighborhood base of the origin, then the family <math>\left\{\Delta(N) : N \in \mathcal{N}(0)\right\}</math> is a base for this uniformity.
}}

This section is dedicated to explaining the precise meanings of the terms involved in this uniqueness statement.

===Uniform spaces and translation-invariant uniformities===
{{Main|Uniform space}}

For any subsets <math>\Phi, \Psi \subseteq X \times X,</math> let{{sfn|Schaefer|Wolff|1999|pp=1-11}}
<math display=block>\Phi^{\operatorname{op}} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{(y, x) ~:~ (x, y) \in \Phi\}</math>
and let
<math display=block>\begin{alignat}{4}
\Phi \circ \Psi ~&~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left\{(x, z) : \text{ there exists } y \in X \text{ such that } (x, y) \in \Psi \text{ and } (y, z) \in \Phi\right\} \\
&=~ \bigcup_{y \in X} \{(x, z) ~:~ (x, y) \in \Psi \text{ and } (y, z) \in \Phi\}
\end{alignat}</math>
A non-empty family <math>\mathcal{B} \subseteq \wp(X \times X)</math> is called a '''{{visible anchor|base of entourages}}''' or a '''{{visible anchor|fundamental system of entourages}}''' if <math>\mathcal{B}</math> is a prefilter on <math>X \times X</math> satisfying all of the following conditions:
<ol>
<li>Every set in <math>\mathcal{B}</math> contains the diagonal of <math>X</math> as a subset; that is, <math>\Delta_X ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{(x, x) : x \in X\} \subseteq \Phi</math> for every <math>\Phi \in \mathcal{B}.</math> Said differently, the prefilter <math>\mathcal{B}</math> is {{em|fixed}} on <math>\Delta_X.</math></li>
<li>For every <math>\Omega \in \mathcal{B}</math> there exists some <math>\Phi \in \mathcal{B}</math> such that <math>\Phi \circ \Phi \subseteq \Omega.</math></li>
<li>For every <math>\Omega \in \mathcal{B}</math> there exists some <math>\Phi \in \mathcal{B}</math> such that <math>\Phi \subseteq \Omega^{\operatorname{op}} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{(y, x) : (x, y) \in \Omega\}.</math></li>
</ol>

A '''{{visible anchor|uniformity}}''' or '''{{visible anchor|uniform structure}}''' on <math>X</math> is a [[Filter (set theory)|filter]] <math>\mathcal{U}</math> on <math>X \times X</math> that is generated by some base of entourages <math>\mathcal{B},</math> in which case we say that <math>\mathcal{B}</math> is a '''base of entourages {{em|for <math>\mathcal{U}.</math>}}'''

For a commutative additive group <math>X,</math> a '''{{visible anchor|translation-invariant fundamental system of entourages}}'''{{sfn|Schaefer|Wolff|1999|pp=12-19}} is a fundamental system of entourages <math>\mathcal{B}</math> such that for every <math>\Phi \in \mathcal{B},</math> <math>(x, y) \in \Phi</math> if and only if <math>(x + z, y + z) \in \Phi</math> for all <math>x, y, z \in X.</math> A uniformity <math>\mathcal{B}</math> is called a '''{{visible anchor|translation-invariant uniformity}}'''{{sfn|Schaefer|Wolff|1999|pp=12-19}} if it has a base of entourages that is translation-invariant. 
The canonical uniformity on any TVS is translation-invariant.{{sfn|Schaefer|Wolff|1999|pp=12-19}}

The binary operator <math>\;\circ\;</math> satisfies all of the following:
<ul>
<li><math>(\Phi \circ \Psi)^{\operatorname{op}} = \Psi^{\operatorname{op}} \circ \Phi^{\operatorname{op}}.</math></li>
<li>If <math>\Phi \subseteq \Phi_2</math> and <math>\Psi \subseteq \Psi_2</math> then <math>\Phi \circ \Psi \subseteq \Phi_2 \circ \Psi_2.</math></li>
<li>Associativity: <math>\Phi \circ (\Psi \circ \Omega) = (\Phi \circ \Psi) \circ \Omega.</math></li>
<li>Identity: <math>\Phi \circ \Delta_X = \Phi = \Delta_X \circ \Phi.</math></li>
<li>Zero: <math>\Phi \circ \varnothing = \varnothing = \varnothing \circ \Phi</math></li>
</ul>

'''Symmetric entourages'''

Call a subset <math>\Phi \subseteq X \times X</math> '''symmetric''' if <math>\Phi = \Phi^{\operatorname{op}},</math> which is equivalent to <math>\Phi^{\operatorname{op}} \subseteq \Phi.</math> 
This equivalence follows from the identity <math>\left(\Phi^{\operatorname{op}}\right)^{\operatorname{op}} = \Phi</math> and the fact that if <math>\Psi \subseteq X \times X,</math> then <math>\Phi \subseteq \Psi</math> if and only if <math>\Phi^{\operatorname{op}} \subseteq \Psi^{\operatorname{op}}.</math> 
For example, the set <math>\Phi^{\operatorname{op}} \cap \Phi</math> is always symmetric for every <math>\Phi \subseteq X \times X.</math> 
And because <math>(\Phi \cap \Psi)^{\operatorname{op}} = \Phi^{\operatorname{op}} \cap \Psi^{\operatorname{op}},</math> if <math>\Phi</math> and <math>\Psi</math> are symmetric then so is <math>\Phi \cap \Psi.</math> 
<!-- <math>R \cap S \neq \varnothing</math> if and only if <math>\Delta_X \cap (R \times S) \neq \varnothing.</math> -->

===Topology generated by a uniformity===

'''Relatives'''

Let <math>\Phi \subseteq X \times X</math> be arbitrary and let <math>\operatorname{Pr}_1, \operatorname{Pr}_2 : X \times X \to X</math> be the canonical projections onto the first and second coordinates, respectively.

For any <math>S \subseteq X,</math> define 
<math display=block>S \cdot \Phi ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{y \in X : \Phi \cap (S \times \{x\}) \neq \varnothing\} ~=~ \operatorname{Pr}_2 (\Phi \cap (S \times X))</math>
<math display=block>\Phi \cdot S ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{x \in X : \Phi \cap (\{x\} \times S) \neq \varnothing\} ~=~ \operatorname{Pr}_1 (\Phi \cap (X \times S)) = S \cdot \left(\Phi^{\operatorname{op}}\right)</math>
where <math>\Phi \cdot S</math> (respectively, <math>S \cdot \Phi</math>) is called the set of '''left''' (respectively, '''right''') '''<math>\Phi</math>-relatives''' of (points in) <math>S.</math> 
Denote the special case where <math>S = \{p\}</math> is a singleton set for some <math>p \in X</math> by:
<math display=block>p \cdot \Phi ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{p\} \cdot \Phi ~=~ \{y \in X : (p, y) \in \Phi\}</math>
<math display=block>\Phi \cdot p ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \Phi \cdot \{p\} ~=~ \{x \in X : (x, p) \in \Phi\} ~=~ p \cdot \left(\Phi^{\operatorname{op}}\right)</math>
If <math>\Phi, \Psi \subseteq X \times X</math> then
<math display=inline>(\Phi \circ \Psi) \cdot S = \Phi \cdot (\Psi \cdot S).</math> 
Moreover, <math>\,\cdot\,</math> [[Distributive property|right distributes over]] both unions and intersections, meaning that if <math>R, S \subseteq X</math> then <math>(R \cup S) \cdot \Phi ~=~ (R \cdot \Phi) \cup (S \cdot \Phi)</math> and <math>(R \cap S) \cdot \Phi ~\subseteq~ (R \cdot \Phi) \cap (S \cdot \Phi).</math>

'''Neighborhoods and open sets'''

Two points <math>x</math> and <math>y</math> are '''<math>\Phi</math>-close''' if <math>(x, y) \in \Phi</math> and a subset <math>S \subseteq X</math> is called '''<math>\Phi</math>-small''' if <math>S \times S \subseteq \Phi.</math>

Let <math>\mathcal{B} \subseteq \wp(X \times X)</math> be a base of entourages on <math>X.</math> The '''{{visible anchor|neighborhood prefilter}}''' at a point <math>p \in X</math> and, respectively, on a subset <math>S \subseteq X</math> are the [[Family of sets|families of sets]]: 
<math display=block>\mathcal{B} \cdot p ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \mathcal{B} \cdot \{p\} = \{\Phi \cdot p : \Phi \in \mathcal{B}\} \qquad \text{ and } \qquad \mathcal{B} \cdot S ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{\Phi \cdot S : \Phi \in \mathcal{B}\}</math>
and the filters on <math>X</math> that each generates is known as the '''{{visible anchor|neighborhood filter}}''' of <math>p</math> (respectively, of <math>S</math>). 
Assign to every <math>x \in X</math> the neighborhood prefilter 
<math display=block>\mathcal{B} \cdot x ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{\Phi \cdot x : \Phi \in \mathcal{B}\}</math>
and use the [[Topological space#Neighborhood definition|neighborhood definition of "open set"]] to obtain a topology on <math>X</math> called the '''topology induced by <math>\mathcal{B}</math>''' or the '''{{visible anchor|induced topology}}'''. 
Explicitly, a subset <math>U \subseteq X</math> is open in this topology if and only if for every <math>u \in U</math> there exists some <math>N \in \mathcal{B} \cdot u</math> such that <math>N \subseteq U;</math> that is, <math>U</math> is open if and only if for every <math>u \in U</math> there exists some <math>\Phi \in \mathcal{B}</math> such that <math>\Phi \cdot u ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{x \in X : (x, u) \in \Phi\} \subseteq U.</math>

The closure of a subset <math>S \subseteq X</math> in this topology is: 
<math display=block>\operatorname{cl}_X S = \bigcap_{\Phi \in \mathcal{B}} (\Phi \cdot S) = \bigcap_{\Phi \in \mathcal{B}} (S \cdot \Phi).</math>

'''Cauchy prefilters and complete uniformities'''

A prefilter <math>\mathcal{F} \subseteq \wp(X)</math> on a uniform space <math>X</math> with uniformity <math>\mathcal{U}</math> is called a '''Cauchy prefilter''' if for every entourage <math>N \in \mathcal{U},</math> there exists some <math>F \in \mathcal{F}</math> such that <math>F \times F \subseteq N.</math>

A uniform space <math>(X, \mathcal{U})</math> is called a '''{{visible anchor|complete uniform space}}''' (respectively, a '''{{visible anchor|sequentially complete uniform space}}''') if every Cauchy prefilter (respectively, every elementary Cauchy prefilter) on <math>X</math> converges to at least one point of <math>X</math> when <math>X</math> is endowed with the topology induced by <math>\mathcal{U}.</math>

'''Case of a topological vector space'''

If <math>(X, \tau)</math> is a [[Topological vector space|topological vector space]] then for any <math>S \subseteq X</math> and <math>x \in X,</math> 
<math display=block>\Delta_X(N) \cdot S = S + N \qquad \text{ and } \qquad \Delta_X(N) \cdot x = x + N,</math>
and the topology induced on <math>X</math> by the canonical uniformity is the same as the topology that <math>X</math> started with (that is, it is <math>\tau</math>).

==Uniform continuity==

Let <math>X</math> and <math>Y</math> be TVSs, <math>D \subseteq X,</math> and <math>f : D \to Y</math> be a map. Then <math>f : D \to Y</math> is '''{{em|uniformly continuous}}''' if for every neighborhood <math>U</math> of the origin in <math>X,</math> there exists a neighborhood <math>V</math> of the origin in <math>Y</math> such that for all <math>x, y \in D,</math> if <math>y - x \in U</math> then <math>f(y) - f(x) \in V.</math>

Suppose that <math>f : D \to Y</math> is uniformly continuous. If <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> is a Cauchy net in <math>D</math> then <math>f \circ x_{\bull} = \left(f\left(x_i\right)\right)_{i \in I}</math> is a Cauchy net in <math>Y.</math> 
If <math>\mathcal{B}</math> is a Cauchy prefilter in <math>D</math> (meaning that <math>\mathcal{B}</math> is a family of subsets of <math>D</math> that is Cauchy in <math>X</math>) then <math>f\left(\mathcal{B}\right)</math> is a Cauchy prefilter in <math>Y.</math> However, if <math>\mathcal{B}</math> is a Cauchy filter on <math>D</math> then although <math>f\left(\mathcal{B}\right)</math> will be a Cauchy {{em|pre}}filter, it will be a Cauchy filter in <math>Y</math> if and only if <math>f : D \to Y</math> is surjective.

==TVS completeness vs completeness of (pseudo)metrics==

===Preliminaries: Complete pseudometric spaces===

{{Main|Complete metric space|Pseudometric space|Cauchy sequence}}

We review the basic notions related to the general theory of complete pseudometric spaces. 
Recall that every [[Metric (mathematics)|metric]] is a pseudometric and that a pseudometric <math>p</math> is a metric if and only if <math>p(x, y) = 0</math> implies <math>x = y.</math> Thus every [[Metric space|metric space]] is a [[Pseudometric space|pseudometric space]] and a pseudometric space <math>(X, p)</math> is a metric space if and only if <math>p</math> is a metric.

If <math>S</math> is a subset of a [[Pseudometric space|pseudometric space]] <math>(X, d)</math> then the '''[[Diameter|diameter]]''' of <math>S</math> is defined to be 
<math display=block>\operatorname{diam}(S) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \sup_{} \{d(s, t) : s, t \in S\}.</math>

A prefilter <math>\mathcal{B}</math> on a pseudometric space <math>(X, d)</math> is called a '''<math>d</math>-Cauchy prefilter''' or simply a '''Cauchy prefilter''' if for each [[Real number|real]] <math>r > 0,</math> there is some <math>B \in \mathcal{B}</math> such that the diameter of <math>B</math> is less than <math>r.</math>

Suppose <math>(X, d)</math> is a pseudometric space. A [[Net (mathematics)|net]] <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> in <math>X</math> is called a '''<math>d</math>-Cauchy net''' or simply a '''Cauchy net''' if <math>\operatorname{Tails}\left(x_{\bull}\right)</math> is a Cauchy prefilter, which happens if and only if

:for every <math>r > 0</math> there is some <math>i \in I</math> such that if <math>j, k \in I</math> with <math>j \geq i</math> and <math>k \geq i</math> then <math>d\left(x_j, x_k\right) < r</math>

or equivalently, if and only if <math>\left(d\left(x_j, x_k\right)\right)_{(i,j) \in I \times I} \to 0</math> in <math>\R.</math> This is analogous to the following characterization of the converge of <math>x_{\bull}</math> to a point: if <math>x \in X,</math> then <math>x_{\bull} \to x</math> in <math>(X, d)</math> if and only if <math>\left(x_i, x\right)_{i \in I} \to 0</math> in <math>\R.</math>

A '''[[Cauchy sequence]]''' is a sequence that is also a Cauchy net.<ref group=note>Every sequence is also a net.</ref>

Every pseudometric <math>p</math> on a set <math>X</math> induces the usual canonical topology on <math>X,</math> which we'll denote by <math>\tau_p</math>; it also induces a canonical [[Uniform space|uniformity]] on <math>X,</math> which we'll denote by <math>\mathcal{U}_p.</math> The topology on <math>X</math> induced by the uniformity <math>\mathcal{U}_p</math> is equal to <math>\tau_p.</math> A net <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> in <math>X</math> is Cauchy with respect to <math>p</math> if and only if it is Cauchy with respect to the uniformity <math>\mathcal{U}_p.</math> 
The pseudometric space <math>(X, p)</math> is a complete (resp. a sequentially complete) pseudometric space if and only if <math>\left(X, \mathcal{U}_p\right)</math> is a complete (resp. a sequentially complete) uniform space. 
Moreover, the pseudometric space <math>(X, p)</math> (resp. the uniform space <math>\left(X, \mathcal{U}_p\right)</math>) is complete if and only if it is sequentially complete.

A pseudometric space <math>(X, d)</math> (for example, a [[Metric space|metric space]]) is called '''complete''' and <math>d</math> is called a '''complete pseudometric''' if any of the following equivalent conditions hold:
<ol>
<li>Every Cauchy prefilter on <math>X</math> converges to at least one point of <math>X.</math></li>
<li>The above statement but with the word "prefilter" replaced by "filter."</li>
<li>Every Cauchy net in <math>X</math> converges to at least one point of <math>X.</math>
* If <math>d</math> is a metric on <math>X</math> then any limit point is necessarily unique and the same is true for limits of Cauchy prefilters on <math>X.</math></li>
<li>Every Cauchy sequence in <math>X</math> converges to at least one point of <math>X.</math>
* Thus to prove that <math>(X, d)</math> is complete, it suffices to only consider Cauchy sequences in <math>X</math> (and it is not necessary to consider the more general Cauchy nets). </li>
<li>The canonical uniformity on <math>X</math> induced by the pseudometric <math>d</math> is a complete uniformity.</li>
</ol>

And if addition <math>d</math> is a metric then we may add to this list:
<ol start=6>
<li>Every decreasing sequence of closed balls whose diameters shrink to <math>0</math> has non-empty intersection.{{sfn|Narici|Beckenstein|2011|pp=64-66}}</li>
</ol>

===Complete pseudometrics and complete TVSs===

Every [[F-space]], and thus also every [[Fréchet space]], [[Banach space]], and [[Hilbert space]] is a complete TVS. Note that every ''F''-space is a [[Baire space]] but there are normed spaces that are Baire but not Banach.{{sfn|Wilansky|2013|p=29}}

A pseudometric <math>d</math> on a vector space <math>X</math> is said to be a '''{{visible anchor|translation invariant pseudometric}}''' if <math>d(x, y) = d(x + z, y + z)</math> for all vectors <math>x, y, z \in X.</math>

Suppose <math>(X, \tau)</math> is [[Metrizable topological vector space|pseudometrizable TVS]] (for example, a metrizable TVS) and that <math>p</math> is {{em|any}} pseudometric on <math>X</math> such that the topology on <math>X</math> induced by <math>p</math> is equal to <math>\tau.</math> 
If <math>p</math> is translation-invariant, then <math>(X, \tau)</math> is a complete TVS if and only if <math>(X, p)</math> is a complete pseudometric space.{{sfn|Narici|Beckenstein|2011|pp=47-51}} 
If <math>p</math> is {{em|not}} translation-invariant, then may be possible for <math>(X, \tau)</math> to be a complete TVS but <math>(X, p)</math> to {{em|not}} be a complete pseudometric space{{sfn|Narici|Beckenstein|2011|pp=47-51}} (see this footnote<ref group=note>The normed space <math>(\R, | \cdot |)</math> is a Banach space where the absolute value is a norm that induces the usual Euclidean topology on <math>\R.</math> Define a metric <math>D</math> on <math>\R</math> by <math>D(x, y) = \left|\arctan(x) - \arctan(y)\right|</math> for all <math>x, y \in \R,</math> where one may show that <math>D</math> induces the usual Euclidean topology on <math>\R.</math> However, <math>D</math> is not a complete metric since the sequence <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math>defined by <math>x_i = i</math> is a <math>D</math>-Cauchy sequence that does not converge in <math>\R</math> to any point of <math>\R.</math> Note also that this <math>D</math>-Cauchy sequence is not a Cauchy sequence in <math>(\R, | \cdot |)</math> (that is, it is not a Cauchy sequence with respect to the norm <math>| \cdot |</math>).</ref> for an example).{{sfn|Narici|Beckenstein|2011|pp=47-51}}

{{Math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|p=35}}<ref name="Klee Inv metrics">{{Cite journal|last1=Klee|first1=V. L.|title=Invariant metrics in groups (solution of a problem of Banach)|year=1952|journal=Proc. Amer. Math. Soc.|volume=3 |issue=3|pages=484–487|url=https://www.ams.org/journals/proc/1952-003-03/S0002-9939-1952-0047250-4/S0002-9939-1952-0047250-4.pdf|doi = 10.1090/s0002-9939-1952-0047250-4 |doi-access=free}}</ref>|note=Klee|math_statement=
Let <math>d</math> be {{em|any}}<ref group=note>Not assumed to be translation-invariant.</ref> metric on a vector space <math>X</math> such that the topology <math>\tau</math> induced by <math>d</math> on <math>X</math> makes <math>(X, \tau)</math> into a topological vector space. If <math>(X, d)</math> is a complete metric space then <math>(X, \tau)</math> is a complete-TVS.
}}

===Complete norms and equivalent norms===

Two norms on a vector space are called '''[[Norm (mathematics)#Definition|equivalent]]''' if and only if they induce the same topology.<ref name="Conrad Equiv norms">{{cite web |url=https://kconrad.math.uconn.edu/blurbs/gradnumthy/equivnorms.pdf |title=Equivalence of norms |last=Conrad |first=Keith |website=kconrad.math.uconn.edu |publisher= |date= |access-date=September 7, 2020 |quote=}}</ref> If <math>p</math> and <math>q</math> are two equivalent norms on a vector space <math>X</math> then the normed space <math>(X, p)</math> is a [[Banach space]] if and only if <math>(X, q)</math> is a Banach space. 
See this footnote for an example of a continuous norm on a Banach space that is {{em|not}} equivalent to that Banach space's given norm.<ref group=note>Let <math>\left(C([0, 1]), \| \cdot \|_{\infty}\right)</math> denotes the [[Continuous functions on a compact Hausdorff space|Banach space of continuous functions]] with the supremum norm, let <math>X = C([0, 1])</math> where <math>X</math> is given the topology induced by <math>\| \cdot \|_{\infty},</math> and denote the restriction of the [[Lp space|L<sup>1</sup>-norm]] to <math>C([0, 1])</math> by <math>\| \cdot \|_1.</math> Then one may show that <math>\| \cdot \|_1 \leq \| \cdot \|_{\infty}</math> so that the norm <math>\| \cdot \|_1 : X \to \R</math> is a continuous function. However, <math>\| \cdot \|_1</math> is {{em|not}} equivalent to the norm <math>\| \cdot \|_{\infty}</math> and so in particular, <math>\left(C([0, 1]), \| \cdot \|_1\right)</math> is {{em|not}} a Banach space.</ref><ref name="Conrad Equiv norms"/> 
All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.<ref>see Corollary1.4.18, p.32 in {{harvtxt|Megginson|1998}}.</ref> Every Banach space is a complete TVS. A normed space is a Banach space (that is, its canonical norm-induced metric is complete) if and only if it is complete as a topological vector space.

==Completions==

A '''completion'''{{sfn|Narici|Beckenstein|2011|pp=60-61}} of a TVS <math>X</math> is a complete TVS that contains a dense vector subspace that is TVS-isomorphic to <math>X.</math> In other words, it is a complete TVS <math>C</math> into which <math>X</math> can be TVS-embedded as a [[Dense set|dense]] vector subspace. Every TVS-embedding is a uniform embedding.

Every topological vector space has a completion. Moreover, every Hausdorff TVS has a {{em|Hausdorff}} completion, which is necessarily unique [[Up to|up to]] TVS-isomorphism. However, all TVSs, even those that are Hausdorff, (already) complete, and/or metrizable have infinitely many non-Hausdorff completions that are {{em|not}} TVS-isomorphic to one another.

===Examples of completions===

For example, the vector space consisting of scalar-valued [[Simple function|simple function]]s <math>f</math> for which <math>|f|_p < \infty</math> (where this seminorm is defined in the usual way in terms of [[Lebesgue integration]]) becomes a seminormed space when endowed with this seminorm, which in turn makes it into both a [[Pseudometric space|pseudometric space]] and a non-Hausdorff non-complete TVS; any completion of this space is a non-Hausdorff complete seminormed space that when [[Quotient space (linear algebra)|quotiented]] by the closure of its origin (so as to [[Quotient space (topology)|obtain a Hausdorff TVS]]) results in (a space [[Linear map|linearly]] isometrically-isomorphic to) the usual complete Hausdorff <math>L^p</math>-space (endowed with the usual complete <math>\| \cdot \|_p</math> [[Norm (mathematics)|norm]]).

As another example demonstrating the usefulness of completions, the completions of [[Topological tensor product|topological tensor product]]s, such as [[Projective tensor product|projective tensor product]]s or [[Injective tensor product|injective tensor product]]s, of the Banach space <math>\ell^1(S)</math> with a complete Hausdorff locally convex TVS <math>Y</math> results in a complete TVS that is TVS-isomorphic to a "generalized" <math>\ell^1(S; Y)</math>-space consisting <math>Y</math>-valued functions on <math>S</math> (where this "generalized" TVS is defined analogously to original space <math>\ell^1(S)</math> of scalar-valued functions on <math>S</math>). Similarly, the completion of the injective tensor product of the space of scalar-valued <math>C^k</math>-test functions with such a TVS <math>Y</math> is TVS-isomorphic to the analogously defined TVS of [[Differentiable vector-valued functions from Euclidean space|<math>Y</math>-valued <math>C^k</math>]] test functions.

===Non-uniqueness of all completions===

As the example below shows, regardless of whether or not a space is Hausdorff or already complete, every [[Topological vector space|topological vector space]] (TVS) has infinitely many non-isomorphic completions.{{sfn|Narici|Beckenstein|2011|pp=93-113}}

However, every Hausdorff TVS has a {{em|Hausdorff}} completion that is unique up to TVS-isomorphism.{{sfn|Narici|Beckenstein|2011|pp=93-113}} But nevertheless, every Hausdorff TVS still has infinitely many non-isomorphic non-Hausdorff completions.

'''Example''' ('''Non-uniqueness of completions'''):{{sfn|Narici|Beckenstein|2011|pp=60-61}} 
Let <math>C</math> denote any complete TVS and let <math>I</math> denote any TVS endowed with the indiscrete topology, which recall makes <math>I</math> into a complete TVS. 
Since both <math>I</math> and <math>C</math> are complete TVSs, so is their product <math>I \times C.</math> 
If <math>U</math> and <math>V</math> are non-empty open subsets of <math>I</math> and <math>C,</math> respectively, then <math>U = I</math> and <math>(U \times V) \cap (\{0\} \times C) = \{0\} \times V \neq \varnothing,</math> which shows that <math>\{0\} \times C</math> is a dense subspace of <math>I \times C.</math> 
Thus by definition of "completion," <math>I \times C</math> is a completion of <math>\{0\} \times C</math> (it doesn't matter that <math>\{0\} \times C</math> is already complete). 
So by identifying <math>\{0\} \times C</math> with <math>C,</math> if <math>X \subseteq C</math> is a dense vector subspace of <math>C,</math> then <math>X</math> has both <math>C</math> and <math>I \times C</math> as completions.

===Hausdorff completions===

Every Hausdorff TVS has a {{em|Hausdorff}} completion that is unique up to TVS-isomorphism.{{sfn|Narici|Beckenstein|2011|pp=93-113}} But nevertheless, as shown above, every Hausdorff TVS still has infinitely many non-isomorphic non-Hausdorff completions.

{{Math theorem|name=Properties of Hausdorff completions{{sfn|Horváth|1966|pp=139-141}}|math_statement=
Suppose that <math>X</math> and <math>C</math> are Hausdorff TVSs with <math>C</math> complete. Suppose that <math>E : X \to C</math> is a TVS-embedding onto a dense vector subspace of <math>C.</math> Then

:'''Universal property''': for every continuous linear map <math>f : X \to Z</math> into a complete Hausdorff TVS <math>Z,</math> there exists a unique continuous linear map <math>F : C \to Z</math> such that <math>f = F \circ E.</math>

If <math>E_2 : X \to C_2</math> is a TVS embedding onto a dense vector subspace of a complete Hausdorff TVS <math>C_2</math> having the above universal property, then there exists a unique (bijective) TVS-isomorphism <math>I : C \to C_2</math> such that <math>E_2 = I \circ E.</math>
}}

{{Math theorem|name=Corollary{{sfn|Horváth|1966|pp=139-141}}|math_statement=
Suppose <math>C</math> is a complete Hausdorff TVS and <math>X</math> is a dense vector subspace of <math>C.</math> Then every continuous linear map <math>f : X \to Z</math> into a complete Hausdorff TVS <math>Z</math> has a unique continuous linear extension to a map <math>C \to Z.</math>
}}

'''Existence of Hausdorff completions'''


A Cauchy filter <math>\mathcal{B}</math> on a TVS <math>X</math> is called a '''{{visible anchor|minimal Cauchy filter}}'''{{sfn|Horváth|1966|pp=139-141}} if there does {{em|not}} exist a Cauchy filter on <math>X</math> that is strictly coarser than <math>\mathcal{B}</math> (that is, "strictly coarser than <math>\mathcal{B}</math>" means contained as a proper subset of <math>\mathcal{B}</math>).

If <math>\mathcal{B}</math> is a Cauchy filter on <math>X</math> then the filter generated by the following prefilter:
<math display=block>\left\{B + N ~:~ B \in \mathcal{B} \text{ and } N \text{ is a neighborhood of } 0 \text{ in } X\right\}</math>
is the unique minimal Cauchy filter on <math>X</math> that is contained as a subset of <math>\mathcal{B}.</math>{{sfn|Horváth|1966|pp=139-141}} 
In particular, for any <math>x \in X,</math> the neighborhood filter at <math>x</math> is a minimal Cauchy filter.

Let <math>\mathbb{M}</math> be the set of all minimal Cauchy filters on <math>X</math> and let <math>E : X \rarr \mathbb{M}</math> be the map defined by sending <math>x \in X</math> to the neighborhood filter of <math>x</math> in <math>X.</math> 
Endow <math>\mathbb{M}</math> with the following vector space structure: 
Given <math>\mathcal{B}, \mathcal{C} \in \mathbb{M}</math> and a scalar <math>s,</math> let <math>\mathcal{B} + \mathcal{C}</math> (resp. <math>s \mathcal{B}</math>) denote the unique minimal Cauchy filter contained in the filter generated by <math>\left\{B + C : B \in \mathcal{B}, C \in \mathcal{C}\right\}</math> (resp. <math>\{s B : B \in \mathcal{B}\}</math>).

For every [[Balanced set|balanced]] neighborhood <math>N</math> of the origin in <math>X,</math> let
<math display=block>\mathbb{U}(N) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left\{\mathcal{B} \in \mathbb{M} ~:~ \text{ there exist } B \in \mathcal{B} \text{ and a neighborhood } V \text{ of the origin in } X \text{ such that } B + V \subseteq N\right\}</math>

If <math>X</math> is Hausdorff then the collection of all sets <math>\mathbb{U}(N),</math> as <math>N</math> ranges over all balanced neighborhoods of the origin in <math>X,</math> forms a vector topology on <math>\mathbb{M}</math> making <math>\mathbb{M}</math> into a complete Hausdorff TVS. Moreover, the map <math>E : X \rarr \mathbb{M}</math> is a TVS-embedding onto a dense vector subspace of <math>\mathbb{M}.</math>{{sfn|Horváth|1966|pp=139-141}}

If <math>X</math> is a [[Metrizable topological vector space|metrizable TVS]] then a Hausdorff completion of <math>X</math> can be constructed using equivalence classes of Cauchy sequences instead of minimal Cauchy filters.

===Non-Hausdorff completions===

This subsection details how every non-Hausdorff TVS <math>X</math> can be TVS-embedded onto a dense vector subspace of a complete TVS. The proof that every Hausdorff TVS has a Hausdorff completion is widely available and so this fact will be used (without proof) to show that every non-Hausdorff TVS also has a completion. These details are sometimes useful for extending results from Hausdorff TVSs to non-Hausdorff TVSs.

Let <math>I = \operatorname{cl} \{0\}</math> denote the closure of the origin in <math>X,</math> where <math>I</math> is endowed with its subspace topology induced by <math>X</math> (so that <math>I</math> has the indiscrete topology). 
Since <math>I</math> has the trivial topology, it is easily shown that every vector subspace of <math>X</math> that is an algebraic complement of <math>I</math> in <math>X</math> is necessarily a [[Complemented subspace|topological complement]] of <math>I</math> in <math>X.</math>{{sfn|Wilansky|2013|p=63}}{{sfn|Schaefer|Wolff|1999|pp=12-35}} 
Let <math>H</math> denote any topological complement of <math>I</math> in <math>X,</math> which is necessarily a Hausdorff TVS (since it is TVS-isomorphic to the quotient TVS <math>X / I</math><ref group=note>This particular quotient map <math>q : X \to X / I</math> is in fact also a closed map.</ref>). 
Since <math>X</math> is the [[Complemented subspace|topological direct sum]] of <math>I</math> and <math>H</math> (which means that <math>X = I \oplus H</math> in the category of TVSs), the canonical map 
<math display=block>I \times H \to I \oplus H = X \quad \text{ given by } \quad (x, y) \mapsto x + y</math>
is a TVS-isomorphism.{{sfn|Schaefer|Wolff|1999|pp=12-35}} 
Let <math>A ~:~ X = I \oplus H ~\to~ I \times H</math> denote the inverse of this canonical map. (As a side note, it follows that every open and every closed subset <math>U</math> of <math>X</math> satisfies <math>U = I + U.</math><ref group=proof>Let <math>W</math> be a neighborhood of the origin in <math>X.</math> Since <math>A(W)</math> is a neighborhood of <math>0</math> in <math>I \times H,</math> there exists an open (resp. closed) neighborhood <math>V</math> of <math>0</math> in <math>H</math> such that <math>I \times V \subseteq A(W)</math> is a neighborhood of the origin. Clearly, <math>V</math> is open (resp. closed) if and only if <math>I \times V</math> is open (resp. closed). Let <math>U = I + V</math> so that <math>A(U) = I \times V \subseteq A(W)</math> where <math>U</math> is open (resp. closed) if and only if <math>V</math> is open (resp. closed).</ref>)

The Hausdorff TVS <math>H</math> can be TVS-embedded, say via the map <math>\operatorname{In}_H : H \to C,</math> onto a dense vector subspace of its completion <math>C.</math> 
Since <math>I</math> and <math>C</math> are complete, so is their product <math>I \times C.</math> 
Let <math>\operatorname{Id}_I : I \to I</math> denote the identity map and observe that the product map <math>\operatorname{Id}_I \times \operatorname{In}_H : I \times H \to I \times C</math> is a TVS-embedding whose image is dense in <math>I \times C.</math> 
Define the map<ref group=note>Explicitly, this map is defined as follows: for each <math>x \in X,</math> let <math>(i, h) = A(x)</math> and so that <math>B(x) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(i, \operatorname{In}_H h\right).</math> Then <math>B(i + h) = \left(i, \operatorname{In}_H h\right)</math> holds for all <math>i \in I</math> and <math>h \in H.</math></ref>
<math display=block>B : X = I \oplus H \to I \times C \quad \text{ by } \quad B ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(\operatorname{Id}_I \times \operatorname{In}_H\right) \circ A</math>
which is a TVS-embedding of <math>X = I \oplus H</math> onto a dense vector subspace of the complete TVS <math>I \times C.</math> 
Moreover, observe that the closure of the origin in <math>I \times C</math> is equal to <math>I \times \{0\},</math> and that <math>I \times \{0\}</math> and <math>\{0\} \times C</math> are topological complements in <math>I \times C.</math>

To summarize,{{sfn|Schaefer|Wolff|1999|pp=12-35}} given any algebraic (and thus topological) complement <math>H</math> of <math>I ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \operatorname{cl} \{0\}</math> in <math>X</math> and given any completion <math>C</math> of the Hausdorff TVS <math>H</math> such that <math>H \subseteq C,</math> then the natural inclusion<ref name=note>where for all <math>i \in I</math> and <math>h \in H,</math> <math>\operatorname{In}_H(i + h) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ i + h.</math></ref>
<math display=block>\operatorname{In}_H : X = I \oplus H \to I \oplus C</math>
is a well-defined TVS-embedding of <math>X</math> onto a dense vector subspace of the complete TVS <math>I \oplus C</math> where moreover, 
<math display=block>X = I \oplus H \subseteq I \oplus C \cong I \times C.</math>

===Topology of a completion===

{{Math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|pp=12-19}}{{sfn|Schaefer|Wolff|1999|pp=36-72}}|note=Topology of a completion|math_statement=
Let <math>C</math> be a complete TVS and let <math>X</math> be a dense vector subspace of <math>X.</math> 
If <math>\mathcal{N}_X(0)</math> is any neighborhood base of the origin in <math>X</math> then the set
<math display=block>\mathcal{N}_X(0) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left\{\operatorname{cl}_C N ~:~ N \in \mathcal{N}_X (0)\right\}</math>
is a neighborhood of the origin in the completion <math>C</math> of <math>X.</math>

If <math>X</math> is locally convex and <math>\mathcal{P}</math> is a family of continuous seminorms on <math>X</math> that generate the topology of <math>X,</math> then the family of all continuous extensions to <math>C</math> of all members of <math>\mathcal{P}</math> is a generating family of seminorms for <math>C.</math>
}}

Said differently, if <math>C</math> is a completion of a TVS <math>X</math> with <math>X \subseteq C</math> and if <math>\mathcal{N}</math> is a neighborhood base of the origin in <math>X,</math> then the family of sets 
<math display=block>\left\{\operatorname{cl}_C N ~:~ N \in \mathcal{N}\right\}</math>
is a neighborhood basis at the origin in <math>C.</math>{{sfn|Narici|Beckenstein|2011|pp=47-66}}

{{Math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|pp=73−121}}|note=Completions of quotients|math_statement=
Let <math>M</math> be a [[Metrizable topological vector space|metrizable topological vector space]] and let <math>N</math> be a closed vector subspace of <math>M.</math> Suppose that <math>C</math> is a completion of <math>M.</math> Then the completion of <math>M / N</math> is TVS-isomorphic to <math>C / \operatorname{cl}_C N.</math> If in addition <math>M</math> is a normed space, then this TVS-isomorphism is also an isometry.
}}

'''Grothendieck's Completeness Theorem'''


Let <math>\mathcal{E}</math> denote the {{em|{{visible anchor|equicontinuous compactology}}}} on the continuous dual space <math>X^\prime,</math> which by definition consists of all [[Equicontinuity|equicontinuous]] weak-* closed and weak-* [[Bounded set (topological vector space)|bounded]] [[Absolutely convex set|absolutely convex subsets]] of <math>X^\prime</math>{{sfn|Jarchow|1981|pp=151,157}} (which are necessarily weak-* compact subsets of <math>X^\prime</math>). Assume that every <math>E^\prime \in \mathcal{E}</math> is endowed with the weak-* topology. 
A [[Filter (set theory)|filter]] <math>\mathcal{B}</math> on <math>X^\prime</math> is said to {{em|{{visible anchor|converge continuously}}}} to <math>x^\prime \in X^\prime</math> if there exists some <math>E^\prime \in \mathcal{E} \cap \mathcal{B}</math> containing <math>x^\prime</math> (that is, <math>x^\prime \in E^\prime</math>) such that the trace of <math>\mathcal{B}</math> on <math>E^\prime,</math> which is the family <math>\mathcal{B}\big\vert_{E^\prime} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left\{B \cap E^\prime : B \in \mathcal{B}\right\},</math> converges to <math>x^\prime</math> in <math>E^\prime</math> (that is, if <math>\mathcal{B}\big\vert_{E^\prime} \to x^\prime</math> in the given weak-* topology).{{sfn|Jarchow|1981|pp=175−178}} 
The filter <math>\mathcal{B}</math> converges continuously to <math>x^\prime</math> if and only if <math>\mathcal{B} - x^\prime</math> converges continuously to the origin, which happens if and only if for every <math>x \in X,</math> the filter <math>\langle \mathcal{B}, x + \mathcal{N} \rangle \to \langle x^\prime, x\rangle</math> in the scalar field (which is <math>\R</math> or <math>\Complex</math>) where <math>\mathcal{N}</math> denotes any neighborhood basis at the origin in <math>X,</math> <math>\langle \cdot, \cdot \rangle</math> denotes the [[Dual system|duality pairing]], and <math>\langle \mathcal{B}, x + \mathcal{N} \rangle</math> denotes the filter generated by <math>\{\langle B, x + N \rangle ~:~ B \in \mathcal{B}, N \in \mathcal{N}\}.</math>{{sfn|Jarchow|1981|pp=175−178}} 
A map <math>f : X^\prime \to T</math> into a topological space (such as <math>\R</math> or <math>\Complex</math>) is said to be {{em|{{visible anchor|γ-continuous|text=<math>\gamma</math>-continuous}}}} if whenever a filter <math>\mathcal{B}</math> on <math>X^\prime</math> converges continuously to <math>x^\prime \in X^\prime,</math> then <math>f(\mathcal{B}) \to f\left(x^\prime\right).</math>{{sfn|Jarchow|1981|pp=175−178}}

{{Math theorem|name={{visible anchor|Grothendieck's Completeness Theorem}}{{sfn|Jarchow|1981|pp=175−178}}|note=|math_statement=
If <math>X</math> is a Hausdorff topological vector space then its completion is linearly isomorphic to the set of all <math>\gamma</math>-continuous linear functions on <math>X^\prime.</math>
}}

===Properties preserved by completions===

If a TVS <math>X</math> has any of the following properties then so does its completion:
<ul>
<li>[[Hausdorff space|Hausdorff]]</li>
<li>[[Locally convex topological vector space|Locally convex]]</li>
<li>[[Metrizable topological vector space|Pseudometrizable]]{{sfn|Narici|Beckenstein|2011|pp=93-113}}</li>
<li>[[Metrizable topological vector space|Metrizable]]{{sfn|Narici|Beckenstein|2011|pp=93-113}}</li>
<li>Seminormable</li>
<li>[[Metrizable topological vector space|Normable]]
* Moreover, if <math>(X, p)</math> is a normed space, then the completion can be chosen to be a Banach space <math>(B, q)</math> such that the TVS-embedding of <math>(X, p)</math> into <math>(B, q)</math> is an isometry.</li>
<li>[[Inner product space|Hausdorff pre-Hilbert]]. That is, a TVS induced by an inner product.{{sfn|Trèves|2006|pp=112-125}}</li>
<li>[[Nuclear space|Nuclear]]{{sfn|Schaefer|Wolff|1999|pp=73-121}}</li>
<li>[[Barrelled space|Barrelled]]{{sfn|Schaefer|Wolff|1999|pp=68-72}}</li>
<li>[[Mackey space|Mackey]]{{sfn|Schaefer|Wolff|1999|pp=122-202}}</li>
<li>[[DF-space]]{{sfn|Schaefer|Wolff|1999|pp=190-202}}</li>
</ul>

'''Completions of Hilbert spaces'''

Every inner product space <math>\left(H, \langle \cdot, \cdot \rangle\right)</math> has a completion <math>\left(\overline{H}, \langle \cdot, \cdot \rangle_{\overline{H}}\right)</math> that is a Hilbert space, where the inner product <math>\langle \cdot, \cdot \rangle_{\overline{H}}</math> is the unique continuous extension to <math>\overline{H}</math> of the original inner product <math>\langle \cdot, \cdot \rangle.</math> The norm induced by <math>\left(\overline{H}, \langle \cdot, \cdot \rangle_{\overline{H}}\right)</math> is also the unique continuous extension to <math>\overline{H}</math> of the norm induced by <math>\langle \cdot, \cdot \rangle.</math>{{sfn|Trèves|2006|pp=112-125}}{{sfn|Schaefer|Wolff|1999|pp=36-72}}

'''Other preserved properties'''

If <math>X</math> is a [[Hausdorff space|Hausdorff]] TVS, then the continuous dual space of <math>X</math> is identical to the continuous dual space of the completion of <math>X.</math>{{sfn|Narici|Beckenstein| 2011|pp=225–273}} The completion of a locally convex [[Bornological space|bornological space]] is a [[Barrelled space|barrelled space]].{{sfn|Schaefer|Wolff|1999|pp=68-72}} If <math>X</math> and <math>Y</math> are [[DF-space]]s then the [[Projective tensor product|projective tensor product]], as well as its completion, of these spaces is a DF-space.{{sfn|Schaefer|Wolff|1999|pp=199-202}}

The completion of the [[Projective tensor product|projective tensor product]] of two nuclear spaces is nuclear.{{sfn|Schaefer|Wolff|1999|pp=73-121}} The completion of a nuclear space is TVS-isomorphic with a projective limit of [[Hilbert space]]s.{{sfn|Schaefer|Wolff|1999|pp=73-121}}

If <math>X = Y \oplus Z</math> (meaning that the addition map <math>Y \times Z \to X</math> is a TVS-isomorphism) has a Hausdorff completion <math>C</math> then <math>\left(\operatorname{cl}_C Y\right) + \left(\operatorname{cl}_C Z\right) = C.</math> 
If in addition <math>X</math> is an [[Inner product space|inner product space]] and <math>Y</math> and <math>Z</math> are [[Orthogonal complement|orthogonal complement]]s of each other in <math>X</math> (that is, <math>\langle Y, Z \rangle = \{0\}</math>), then <math>\operatorname{cl}_C Y</math> and <math>\operatorname{cl}_C Z</math> are orthogonal complements in the [[Hilbert space]] <math>C.</math>

===Properties of maps preserved by extensions to a completion===

If <math>f : X \to Y</math> is a nuclear linear operator between two locally convex spaces and if <math>C</math> be a completion of <math>X</math> then <math>f</math> has a unique continuous linear extension to a nuclear linear operator <math>F : C \to Y.</math>{{sfn|Schaefer|Wolff|1999|pp=73-121}}

Let <math>X</math> and <math>Y</math> be two Hausdorff TVSs with <math>Y</math> complete. Let <math>C</math> be a completion of <math>X.</math> Let <math>L(X; Y)</math> denote the vector space of continuous linear operators and let <math>I : L(X; Y) \to L(C; Y)</math> denote the map that sends every <math>f \in L(X; Y)</math>to its unique continuous linear extension on <math>C.</math> Then <math>I : L(X; Y) \to L(C; Y)</math> is a (surjective) vector space isomorphism. Moreover, <math>I : L(X; Y) \to L(C; Y)</math> maps families of equicontinuous subsets onto each other. Suppose that <math>L(X; Y)</math> is endowed with a <math>\mathcal{G}</math>-topology and that <math>\mathcal{H}</math> denotes the closures in <math>C</math> of sets in <math>\mathcal{G}.</math> Then the map <math>I : L_{\mathcal{G}}(X; Y) \to L_{\mathcal{H}}(C; Y)</math> is also a TVS-isomorphism.{{sfn|Schaefer|Wolff|1999|pp=73-121}}

==Examples and sufficient conditions for a complete TVS==

{{Math theorem|name=Theorem|math_statement={{sfn|Schaefer|Wolff|1999|p=35}}
Let <math>d</math> be {{em|any}} (not assumed to be translation-invariant) metric on a vector space <math>X</math> such that the topology <math>\tau</math> induced by <math>d</math> on <math>X</math> makes <math>(X, \tau)</math> into a topological vector space. If <math>(X, d)</math> is a complete metric space then <math>(X, \tau)</math> is a complete-TVS.
}}

<ul>
<li>Any TVS endowed with the [[Trivial topology|trivial topology]] is complete and every one of its subsets is complete. Moreover, every TVS with the trivial topology is compact and hence locally compact. Thus a complete seminormable locally convex and locally compact TVS need not be finite-dimensional if it is not Hausdorff.</li>
<li>An arbitrary product of complete (resp. sequentially complete, quasi-complete) TVSs has that same property. If all spaces are Hausdorff, then the converses are also true.{{sfn|Jarchow|1981|pp=56-73}} A product of Hausdorff completions of a family of (Hausdorff) TVSs is a Hausdorff completion of their product TVS.{{sfn|Jarchow|1981|pp=56-73}} More generally, an arbitrary product of complete subsets of a family of TVSs is a complete subset of the product TVS.{{sfn|Narici|Beckenstein|2011|p=57}}</li>
<li>The projective limit of a projective system of Hausdorff complete (resp. sequentially complete, quasi-complete) TVSs has that same property.{{sfn|Jarchow|1981|pp=56-73}} A projective limit of Hausdorff completions of an inverse system of (Hausdorff) TVSs is a Hausdorff completion of their projective limit.{{sfn|Jarchow|1981|pp=56-73}}</li>
<li>If <math>M</math> is a closed vector subspace of a complete [[Metrizable topological vector space|pseudometrizable TVS]] <math>X,</math> then the quotient space <math>X / M</math> is complete.{{sfn|Narici|Beckenstein|2011|pp=47-66}}</li>
<li>Suppose <math>M</math> is a {{em|complete}} vector subspace of a [[Metrizable topological vector space|metrizable TVS]] <math>X.</math> If the quotient space <math>X / M</math> is complete then so is <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=47-66}}{{sfn|Horváth|1966|pp=129-141}} However, there exists a complete TVS <math>X</math> having a closed vector subspace <math>M</math> such that the quotient TVS <math>X / M</math> is {{em|not}} complete.{{sfn|Horváth|1966|pp=139-141}}</li>
<li>Every [[F-space]], [[Fréchet space]], [[Banach space]], and [[Hilbert space]] is a complete TVS.</li>
<li>Strict [[LF-space]]s and strict [[LB-space]]s are complete.{{sfn|Narici|Beckenstein|2011|pp=441-457}}</li>
<li>Suppose that <math>D</math> is a dense subset of a TVS <math>X.</math> If every Cauchy filter on <math>D</math> converges to some point in <math>X</math> then <math>X</math> is complete.{{sfn|Horváth|1966|pp=129-141}}</li>
<li>The [[Schwartz space]] of smooth functions is complete.</li>
<li>The spaces of [[Distribution (mathematics)|distribution]]s and test functions is complete.</li>
<li>Suppose that <math>X</math> and <math>Y</math> are locally convex TVSs and that the space of continuous linear maps <math>L_b(X; Y)</math> is endowed with the [[Topologies on spaces of linear maps|topology of uniform convergence on bounded subsets]] of <math>X.</math> If <math>X</math> is a [[Bornological space|bornological space]] and if <math>Y</math> is complete then <math>L_b(X; Y)</math> is a complete TVS.{{sfn|Narici|Beckenstein|2011|pp=441-457}} In particular, the strong dual of a [[Bornological space|bornological space]] is complete.{{sfn|Narici|Beckenstein|2011|pp=441-457}} However, it need not be bornological.</li>
<li>Every quasi-complete [[DF-space]] is complete.{{sfn|Schaefer|Wolff|1999|pp=190-202}}</li>
<li>Let <math>\omega</math> and <math>\tau</math> be Hausdorff TVS topologies on a vector space <math>X</math> such that <math>\omega \subseteq \tau.</math> If there exists a prefilter <math>\mathcal{B}</math> such that <math>\mathcal{B}</math> is a neighborhood basis at the origin for <math>(X, \tau)</math> and such that every <math>B \in \mathcal{B}</math> is a complete subset of <math>(X, \omega),</math> then <math>(X, \tau)</math> is a complete TVS.{{sfn|Narici|Beckenstein|2011|pp=48-51}}</li><!-------- Removed: Let <math>\mathcal{N}_{\tau} (0)</math> denote the neighborhood filter at the origin in <math>(X, \tau).</math> ------>
</ul>

==Properties==

===Complete TVSs===

Every TVS has a completion and every Hausdorff TVS has a Hausdorff completion.{{sfn|Narici|Beckenstein|2011|pp=67-113}} 
Every complete TVS is [[Quasi-complete space|quasi-complete space]] and [[Sequentially complete|sequentially complete]].{{sfn|Narici|Beckenstein|2011|pp=155-176}} 
However, the converses of the above implications are generally false.{{sfn|Narici|Beckenstein|2011|pp=155-176}} 
There exists a [[Sequentially complete|sequentially complete]] locally convex TVS that is not [[Quasi-complete space|quasi-complete]].{{sfn|Schaefer|Wolff|1999|pp=190-202}}

If a TVS has a complete neighborhood of the origin then it is complete.{{sfn|Narici|Beckenstein|2011|pp=115-154}} 
Every complete [[Metrizable topological vector space|{{em|pseudo}}metrizable TVS]] is a [[Barrelled space|barrelled space]] and a [[Baire space]] (and thus non-meager).{{sfn|Narici|Beckenstein|2011|pp=371-423}} 
The dimension of a complete metrizable TVS is either finite or uncountable.{{sfn|Schaefer|Wolff|1999|pp=12-35}}

===Cauchy nets and prefilters===

Any neighborhood basis of any point in a TVS is a Cauchy prefilter.

Every convergent net (respectively, prefilter) in a TVS is necessarily a Cauchy net (respectively, a Cauchy prefilter).{{sfn|Narici|Beckenstein|2011|pp=48-51}} 
Any prefilter that is subordinate to (that is, finer than) a Cauchy prefilter is necessarily also a Cauchy prefilter{{sfn|Narici|Beckenstein|2011|pp=48-51}} and any prefilter finer than a Cauchy prefilter is also a Cauchy prefilter. 
The filter associated with a sequence in a TVS is Cauchy if and only if the sequence is a Cauchy sequence. 
Every convergent prefilter is a Cauchy prefilter.

If <math>X</math> is a TVS and if <math>x \in X</math> is a cluster point of a Cauchy net (respectively, Cauchy prefilter), then that Cauchy net (respectively, that Cauchy prefilter) converges to <math>x</math> in <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=47-66}} 
If a Cauchy filter in a TVS has an [[Filters in topology|accumulation point]] <math>x</math> then it converges to <math>x.</math>

Uniformly continuous maps send Cauchy nets to Cauchy nets.{{sfn|Narici|Beckenstein|2011|pp=47-66}} 
A Cauchy sequence in a Hausdorff TVS <math>X,</math> when considered as a set, is not necessarily relatively compact (that is, its closure in <math>X</math> is not necessarily compact<ref group=note>If <math>X</math> is a normable TVS such that for every Cauchy sequence <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty},</math> the closure of <math>S ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{x_1, x_2, \ldots,\}</math> in <math>X</math> is compact (and thus sequentially compact) then this guarantees that there always exist some <math>x \in \operatorname{cl}_X S</math> such that <math>x_{\bull} \to x</math> in <math>X.</math> Thus any normed space with this property is necessarily sequentially complete. Since not all normed spaces are complete, the closure of a Cauchy sequence is not necessarily compact.</ref>) although it is precompact (that is, its closure in the completion of <math>X</math> is compact).

Every Cauchy sequence is a bounded subset but this is not necessarily true of Cauchy net. For example, let <math>\N</math> have it usual order, let <math>\,\leq\,</math> denote any [[Preorder|preorder]] on the non-indiscrete TVS <math>X</math> (that is, <math>X</math> does not have the [[Trivial topology|trivial topology]]; it is also assumed that <math>X \cap \N = \varnothing</math>) and extend these two preorders to the union <math>I ~\stackrel{\scriptscriptstyle\text{def}}{=}~ X \cup \N</math> by declaring that <math>x \leq n</math> holds for every <math>x \in X</math> and <math>n \in \N.</math> 
Let <math>f : I \to X</math> be defined by <math>f(i) = i</math> if <math>i \in X</math> and <math>f(i) = 0</math> otherwise (that is, if <math>i \in \N</math>), which is a net in <math>X</math> since the preordered set <math>(I, \leq)</math> is [[Directed set|directed]] (this preorder on <math>I</math> is also partial order (respectively, a [[Total order|total order]]) if this is true of <math>(X, \leq)</math>). This net <math>f</math> is a Cauchy net in <math>X</math> because it converges to the origin, but the set <math>\{f(i) : i \in I\} = X</math> is not a bounded subset of <math>X</math> (because <math>X</math> does not have the trivial topology).

Suppose that <math>X_{\bull} = \left(X_i\right)_{i \in I}</math> is a family of TVSs and that <math>X</math> denotes the product of these TVSs. Suppose that for every index <math>i,</math> <math>\mathcal{B}_i</math>is a prefilter on <math>X_i.</math> Then the product of this family of prefilters is a Cauchy filter on <math>X</math> if and only if each <math>\mathcal{B}_i</math>is a Cauchy filter on <math>X_i.</math>{{sfn|Horváth|1966|pp=139-141}}

===Maps===

If <math>f : X \to Y</math> is an injective [[Topological homomorphism|topological homomorphism]] from a complete TVS into a Hausdorff TVS then the image of <math>f</math> (that is, <math>f(X)</math>) is a closed subspace of <math>Y.</math>{{sfn|Horváth|1966|pp=129-141}} 
If <math>f : X \to Y</math> is a [[Topological homomorphism|topological homomorphism]] from a complete [[Metrizable topological vector space|metrizable]] TVS into a Hausdorff TVS then the range of <math>f</math> is a closed subspace of <math>Y.</math>{{sfn|Horváth|1966|pp=129-141}} 
If <math>f : X \to Y</math> is a uniformly continuous map between two Hausdorff TVSs then the image under <math>f</math> of a totally bounded subset of <math>X</math> is a totally bounded subset of <math>Y.</math>{{sfn|Horváth|1966|pp=145-149}}

'''Uniformly continuous extensions'''

Suppose that <math>f : D \to Y</math> is a uniformly continuous map from a dense subset <math>D</math> of a TVS <math>X</math> into a complete Hausdorff TVS <math>Y.</math> Then <math>f</math> has a unique uniformly continuous extension to all of <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=47-66}} 
If in addition <math>f</math> is a homomorphism then its unique uniformly continuous extension is also a homomorphism.{{sfn|Narici|Beckenstein|2011|pp=47-66}} 
This remains true if "TVS" is replaced by "commutative topological group."{{sfn|Narici|Beckenstein|2011|pp=47-66}} 
The map <math>f</math> is not required to be a linear map and that <math>D</math> is not required to be a vector subspace of <math>X.</math>

'''Uniformly continuous linear extensions'''

Suppose <math>f : X \to Y</math> be a continuous linear operator between two Hausdorff TVSs. If <math>M</math> is a dense vector subspace of <math>X</math> and if the restriction <math>f\big\vert_M : M \to Y</math> to <math>M</math> is a [[Topological homomorphism|topological homomorphism]] then <math>f : X \to Y</math> is also a topological homomorphism.{{sfn|Schaefer|Wolff|1999|p=116}} So if <math>C</math> and <math>D</math> are Hausdorff completions of <math>X</math> and <math>Y,</math> respectively, and if <math>f : X \to Y</math> is a topological homomorphism, then <math>f</math>'s unique continuous linear extension <math>F : C \to D</math> is a topological homomorphism. (Note that it's possible for <math>f : X \to Y</math> to be surjective but for <math>F : C \to D</math> to {{em|not}} be injective.){{sfn|Schaefer|Wolff|1999|p=116}}

Suppose <math>X</math> and <math>Y</math> are Hausdorff TVSs, <math>M</math> is a dense vector subspace of <math>X,</math> and <math>N</math> is a dense vector subspaces of <math>Y.</math> If <math>M</math> are and <math>N</math> are topologically isomorphic additive subgroups via a topological homomorphism <math>f</math> then the same is true of <math>X</math> and <math>Y</math> via the unique uniformly continuous extension of <math>f</math> (which is also a homeomorphism).{{sfn|Narici|Beckenstein|2011|p=59}}

===Subsets===

'''Complete subsets'''

Every complete subset of a TVS is [[Sequentially complete|sequentially complete]]. 
A complete subset of a Hausdorff TVS <math>X</math> is a closed subset of <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=47-66}}{{sfn|Narici|Beckenstein|2011|
pp=115-154}}

Every compact subset of a TVS is complete (even if the TVS is not Hausdorff or not complete).{{sfn|Narici|Beckenstein|2011|pp=47-66}}{{sfn|Narici|Beckenstein|2011|pp=115-154}} 
Closed subsets of a complete TVS are complete; however, if a TVS <math>X</math> is not complete then <math>X</math> is a closed subset of <math>X</math> that is not complete. 
The empty set is complete subset of every TVS. 
If <math>C</math> is a complete subset of a TVS (the TVS is not necessarily Hausdorff or complete) then any subset of <math>C</math> that is closed in <math>C</math> is complete.{{sfn|Narici|Beckenstein|2011|pp=115-154}}

'''Topological complements'''

If <math>X</math> is a non-normable [[Fréchet space]] on which there exists a continuous norm then <math>X</math> contains a closed vector subspace that has no topological complement.{{sfn|Schaefer|Wolff|1999|pp=190-202}} 
If <math>X</math> is a complete TVS and <math>M</math> is a closed vector subspace of <math>X</math> such that <math>X / M</math> is not complete, then <math>H</math> does {{em|not}} have a topological complement in <math>X.</math>{{sfn|Schaefer|Wolff|1999|pp=190-202}}

'''Subsets of completions'''

Let <math>M</math> be a [[Separable space|separable]] locally convex [[Metrizable topological vector space|metrizable topological vector space]] and let <math>C</math> be its completion. If <math>S</math> is a bounded subset of <math>C</math> then there exists a bounded subset <math>R</math> of <math>X</math> such that <math>S \subseteq \operatorname{cl}_C R.</math>{{sfn|Schaefer|Wolff|1999|pp=190-202}}

'''Relation to compact subsets'''

A subset of a TVS ({{em|not}} assumed to be Hausdorff or complete) is [[Compact space|compact]] if and only if it is complete and totally bounded.{{sfn|Narici|Beckenstein|2011|pp=55-56}}<ref group=proof>Suppose <math>S</math> is compact in <math>X</math> and let <math>\mathcal{C}</math> be a Cauchy filter on <math>S.</math> Let <math>\mathcal{D} = \left\{\operatorname{cl}_S C ~:~ C \in \mathcal{C}\right\}</math> so that <math>\mathcal{D}</math> is a Cauchy filter of closed sets. Since <math>\mathcal{D}</math> has the finite intersection property, there exists some <math>s \in S</math> such that <math>s \operatorname{cl}_S C</math> for all <math>C \in \mathcal{C}</math> so {<math>s \in \operatorname{cl} \mathcal{C}</math> (that is, <math>s</math> is an accumulation point of <math>\mathcal{C}</math>). Since <math>\mathcal{C}</math> is Cauchy, <math>\mathcal{C} \to x</math> in <math>S.</math> Thus <math>S</math> is complete. That <math>S</math> is also totally bounded follows immediately from the compactness of <math>S.</math></ref> 
Thus a closed and totally bounded subset of a complete TVS is compact.{{sfn|Narici|Beckenstein|2011|pp=55-66}}{{sfn|Narici|Beckenstein|2011|pp=47-66}}

In a Hausdorff locally convex TVS, the convex hull of a [[Totally bounded space#Topological vector spaces|precompact]] set is again precompact.{{sfn|Trèves|2006|p=67}} Consequently, in a complete locally convex Hausdorff TVS, the closed convex hull of a compact subset is again compact.{{sfn|Trèves|2006|p=145}}

The convex hull of compact subset of a [[Hilbert space]] is {{em|not}} necessarily closed and so also {{em|not}} necessarily compact. For example, let <math>H</math> be the separable Hilbert space [[Lp 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)</math> be the standard [[Orthonormal basis|orthonormal basis]] (that is <math>1</math> at the <math>n^{\text{th}}</math>-coordinate). The closed set <math>S = \{0\} \cup \left\{\tfrac{1}{n} e_n\right\}</math> is compact but its convex hull <math>\operatorname{co} S</math> is {{em|not}} a closed set because <math>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 sequence <math>z \in \operatorname{co} S</math> is a finite [[Convex combination|convex combination]] of elements of <math>S</math> and so is necessarily <math>0</math> in all but finitely many coordinates, which is not true of <math>h</math>).{{sfn|Aliprantis|Border|2006|p=185}} However, like in all complete Hausdorff locally convex spaces, the {{em|closed}} convex hull <math>K := \overline{\operatorname{co}} S</math> of this compact subset is compact.{{sfn|Trèves|2006|p=145}} The vector subspace <math>X := \operatorname{span} S</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 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 compact subset might {{em|fail}} to be compact (although it will be [[Totally bounded space|precompact/totally bounded]]).

Every complete totally bounded set is relatively compact.{{sfn|Narici|Beckenstein|2011|pp=47-66}} 
If <math>X</math> is any TVS then the quotient map <math>q : X \to X / \operatorname{cl}_X \{0\}</math> is a [[Open and closed maps|closed map]]{{sfn|Narici|Beckenstein|2011|pp=107-112}} and thus <math>S + \operatorname{cl}_X \{0\} \subseteq \operatorname{cl}_X S</math> A subset <math>S</math> of a TVS <math>X</math> is totally bounded if and only if its image under the canonical quotient map <math>q : X \to X / \operatorname{cl}_X \{0\}</math> is totally bounded.{{sfn|Schaefer|Wolff|1999|pp=12-35}} Thus <math>S</math> is totally bounded if and only if <math>S + \operatorname{cl}_X \{0\}</math> is totally bounded. In any TVS, the closure of a totally bounded subset is again totally bounded.{{sfn|Narici|Beckenstein|2011|pp=47-66}} 
In a locally convex space, the convex hull and the [[Absolutely convex set|disked hull]] of a totally bounded set is totally bounded.{{sfn|Narici|Beckenstein|2011|pp=67-113}} If <math>S</math> is a subset of a TVS <math>X</math> such that every sequence in <math>S</math> has a cluster point in <math>S</math> then <math>S</math> is totally bounded.{{sfn|Schaefer|Wolff|1999|pp=12-35}} A subset <math>S</math> of a Hausdorff TVS <math>X</math> is totally bounded if and only if every ultrafilter on <math>S</math> is Cauchy, which happens if and only if it is pre-compact (that is, its closure in the completion of <math>X</math> is compact).{{sfn|Horváth|1966|pp=145-149}}

If <math>S \subseteq X</math> is compact, then <math>\operatorname{cl}_X S = S + \operatorname{cl}_X \{0\}</math> and this set is compact. Thus the closure of a compact set is compact<ref group=note>In general topology, the closure of a compact subset of a non-Hausdorff space may fail to be compact (for example, the [[Particular point topology|particular point topology]] on an infinite set). This result shows that this does not happen in non-Hausdorff TVSs. The proof uses the fact that <math>S</math> is compact (but possibly not closed) and <math>\operatorname{cl}_X \{0\}</math>is both closed and compact so that <math>S + \operatorname{cl}_X \{0\},</math> which is the image of the compact set <math>S + \operatorname{cl}_X \{0\}</math> under the continuous addition map <math>\cdot + \cdot : X \times X \to X,</math> is also compact. Recall also that the sum of a compact set (that is, <math>S</math>) and a closed set is closed so <math>S + \operatorname{cl}_X \{0\}</math> is closed in <math>X.</math></ref> (that is, all compact sets are relatively compact).{{sfn|Narici|Beckenstein|2011|p=156}} Thus the closure of a compact set is compact. Every relatively compact subset of a Hausdorff TVS is totally bounded.{{sfn|Horváth|1966|pp=145-149}}

In a complete locally convex space, the convex hull and the disked hull of a compact set are both compact.{{sfn|Narici|Beckenstein|2011|pp=67-113}} More generally, if <math>K</math> is a compact subset of a locally convex space, then the convex hull <math>\operatorname{co} K</math> (resp. the disked hull <math>\operatorname{cobal} K</math>) is compact if and only if it is complete.{{sfn|Narici|Beckenstein|2011|pp=67-113}} 
Every subset <math>S</math> of <math>\operatorname{cl}_X \{0\}</math> is compact and thus complete.<ref group=proof>Given any open cover of <math>S,</math> pick any open set <math>U</math> from that cover that contains the origin. Since <math>U</math> is a neighborhood of the origin, <math>U</math> contains <math>\operatorname{cl}_X \{0\}</math> and thus contains <math>S.</math></ref> In particular, if <math>X</math> is not Hausdorff then there exist compact complete sets that are not closed.{{sfn|Narici|Beckenstein|2011|pp=47-66}}

==See also==

* {{annotated link|Complete metric space}}
* {{annotated link|Filter (set theory)}}
* {{annotated link|Filters in topology}}
* {{annotated link|Metrizable topological vector space}}
* {{annotated link|Pseudometric space}}
* {{annotated link|Quasi-complete space}}
* {{annotated link|Sequentially complete}}
* {{annotated link|Topological group}}
* {{annotated link|Uniform space}}

==Notes==

{{reflist|group=note}}

'''Proofs'''

{{reflist|group=proof}}

==Citations==

{{reflist}}

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{{Functional analysis}}

[[Category:Functional analysis]]
[[Category:Topological vector spaces]]
[[Category:Uniform spaces]]

{{Sourceattribution|Complete topological vector space}}