Infunctional analysis and related areas ofmathematics, acomplete topological vector space is atopological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point${\displaystyle x}$ towards which they all get closer. The notion of "points that get progressively closer" is made rigorous byCauchy nets orCauchy filters, which are generalizations ofCauchy sequences, while "point${\displaystyle x}$ towards which they all get closer" means that this Cauchynet or filterconverges to${\displaystyle x.}$ The notion of completeness for TVSs uses the theory ofuniform spaces as a framework to generalize the notion ofcompleteness for metric spaces. But unlike metric-completeness, TVS-completeness does not depend on any metric and is defined forall TVSs, including those that are notmetrizable orHausdorff.

Completeness is an extremely important property for a topological vector space to possess. The notions of completeness fornormed spaces andmetrizable TVSs, which are commonly defined in terms ofcompleteness of a particular norm or metric, can both be reduced down to this notion of TVS-completeness – a notion that is independent of any particular norm or metric. Ametrizable topological vector space${\displaystyle X}$ with atranslation invariant metric^{[note 1]}${\displaystyle d}$ is complete as a TVS if and only if${\displaystyle (X,d)}$ is acomplete metric space, which by definition means that every${\displaystyle d}$-Cauchy sequence converges to some point in${\displaystyle X.}$ Prominent examples of complete TVSs that are alsometrizable include allF-spaces and consequently also allFréchet spaces,Banach spaces, andHilbert spaces. Prominent examples of complete TVS that are (typically)not metrizable include strictLF-spaces such as thespace of test functions${\displaystyle C_c^{\mathrm{\infty }}(U)}$ with it canonical LF-topology, thestrong dual space of any non-normableFréchet space, as well as many otherpolar topologies oncontinuous dual space or othertopologies on spaces of linear maps.

Explicitly, atopological vector spaces (TVS) iscomplete if everynet, or equivalently, everyfilter, that isCauchy with respect to the space'scanonicaluniformity necessarily converges to some point. Said differently, a TVS is complete if its canonical uniformity is acomplete uniformity. Thecanonical uniformity on a TVS${\displaystyle (X,\tau )}$ is the unique^{[note 2]} translation-invariantuniformity that induces on${\displaystyle X}$ the topology${\displaystyle \tau .}$ This notion of "TVS-completeness" dependsonly 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 termsmetrics orpseudometrics. Afirst-countable TVS is complete if and only if every Cauchy sequence (or equivalently, everyelementary Cauchy filter) converges to some point.

Every topological vector space${\displaystyle X,}$ even if it is notmetrizable or notHausdorff, has acompletion, which by definition is a complete TVS${\displaystyle C}$ into which${\displaystyle X}$ can beTVS-embedded as adensevector subspace. Moreover, every Hausdorff TVS has aHausdorff completion, which is necessarily uniqueup toTVS-isomorphism. However, as discussed below, all TVSs have infinitely many non-Hausdorff completions that arenot TVS-isomorphic to one another.

This section summarizes the definition of a completetopological vector space (TVS) in terms of bothnets andprefilters. Information about convergence of nets and filters, such as definitions and properties, can be found in the article aboutfilters in topology.

Every topological vector space (TVS) is a commutativetopological group with identity under addition and the canonical uniformity of a TVS is definedentirely in terms of subtraction (and thus addition); scalar multiplication is not involved and no additional structure is needed.

Thediagonal of${\displaystyle X}$ is the set^[1]$${\displaystyle {\mathrm{\Delta }}_X\stackrel{\text{def}}{=}\{(x,x):x\in X\}}$$ and for any${\displaystyle N\subseteq X,}$ thecanonical entourage/vicinity around${\displaystyle N}$ is the setwhere if${\displaystyle 0\in N}$ then${\displaystyle {\mathrm{\Delta }}_X(N)}$ contains the diagonal${\displaystyle {\mathrm{\Delta }}_X(\{0\})={\mathrm{\Delta }}_X.}$

If${\displaystyle N}$ is asymmetric set (that is, if${\displaystyle -N=N}$), then${\displaystyle {\mathrm{\Delta }}_X(N)}$ issymmetric, which by definition means that${\displaystyle {\mathrm{\Delta }}_X(N)={\left({\mathrm{\Delta }}_X(N)\right)}^{\mathrm{op}}}$ holds where${\displaystyle {\left({\mathrm{\Delta }}_X(N)\right)}^{\mathrm{op}}\stackrel{\text{def}}{=}\left\{(y,x):(x,y)\in {\mathrm{\Delta }}_X(N)\right\},}$ and in addition, this symmetric set'scomposition with itself is:

If${\displaystyle \mathcal{L}}$ is any neighborhood basis at the origin in${\displaystyle (X,\tau )}$ then thefamily of subsets of${\displaystyle X\times X:}$${\displaystyle {\mathcal{B}}_{\mathcal{L}}\stackrel{\text{def}}{=}\left\{{\mathrm{\Delta }}_X(N):N\in \mathcal{L}\right\}}$is aprefilter on${\displaystyle X\times X.}$ If${\displaystyle {\mathcal{N}}_{\tau }(0)}$ is theneighborhood filter at the origin in${\displaystyle (X,\tau )}$ then${\displaystyle {\mathcal{B}}_{{\mathcal{N}}_{\tau }(0)}}$ forms abase of entourages for auniform structure on${\displaystyle X}$ that is consideredcanonical.^[2] Explicitly, by definition,thecanonical uniformity on${\displaystyle X}$ induced by${\displaystyle (X,\tau )}$^[2] is thefilter${\displaystyle {\mathcal{U}}_{\tau }}$ on${\displaystyle X\times X}$generated by the above prefilter:$${\displaystyle {\mathcal{U}}_{\tau }\stackrel{\text{def}}{=}{\mathcal{B}}_{{\mathcal{N}}_{\tau }(0)}^{\uparrow }\stackrel{\text{def}}{=}\left\{S\subseteq X\times X:\text{there exists }N\in {\mathcal{N}}_{\tau }(0)\text{ such that }{\mathrm{\Delta }}_X(N)\subseteq S\right\}}$$where${\displaystyle {\mathcal{B}}_{{\mathcal{N}}_{\tau }(0)}^{\uparrow }}$ denotes theupward closure of${\displaystyle {\mathcal{B}}_{{\mathcal{N}}_{\tau }(0)}}$ in${\displaystyle X\times X.}$ The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin. If${\displaystyle \mathcal{L}}$ is any neighborhood basis at the origin in${\displaystyle (X,\tau )}$ then the filter on${\displaystyle X\times X}$ generated by the prefilter${\displaystyle {\mathcal{B}}_{\mathcal{L}}}$ is equal to the canonical uniformity${\displaystyle {\mathcal{U}}_{\tau }}$ induced by${\displaystyle (X,\tau ).}$

The general theory ofuniform spaces has its own definition of a "Cauchy prefilter" and "Cauchy net". For the canonical uniformity on${\displaystyle X,}$ these definitions reduce down to those given below.

Suppose${\displaystyle x_{\bullet }={\left(x_i\right)}_{i\in I}}$ is a net in${\displaystyle X}$ and${\displaystyle y_{\bullet }={\left(y_j\right)}_{j\in J}}$ is a net in${\displaystyle Y.}$ The product${\displaystyle I\times J}$ becomes adirected set by declaring${\displaystyle (i,j)\le \left(i_2,j_2\right)}$ if and only if${\displaystyle i\le i_2}$ and${\displaystyle j\le j_2.}$ Then$${\displaystyle x_{\bullet }\times y_{\bullet }\stackrel{\text{def}}{=}{\left(x_i,y_j\right)}_{(i,j)\in I\times J}}$$denotes the (Cartesian)product net, where in particular$x_{\bullet }\times x_{\bullet }\stackrel{\text{def}}{=}{\left(x_i,x_j\right)}_{(i,j)\in I\times I}.$ If${\displaystyle X=Y}$ then the image of this net under the vector addition map${\displaystyle X\times X\to X}$ denotes thesum of these two nets:^[3]$${\displaystyle x_{\bullet }+y_{\bullet }\stackrel{\text{def}}{=}{\left(x_i+y_j\right)}_{(i,j)\in I\times J}}$$and similarly theirdifference is defined to be the image of the product net under the vector subtraction map${\displaystyle (x,y)\mapsto x-y}$:$${\displaystyle x_{\bullet }-y_{\bullet }\stackrel{\text{def}}{=}{\left(x_i-y_j\right)}_{(i,j)\in I\times J}.}$$ In particular, the notation${\displaystyle x_{\bullet }-x_{\bullet }={\left(x_i\right)}_{i\in I}-{\left(x_i\right)}_{i\in I}}$ denotes the${\displaystyle I^2}$-indexed net${\displaystyle {\left(x_i-x_j\right)}_{(i,j)\in I\times I}}$ and not the${\displaystyle I}$-indexed net${\displaystyle {\left(x_i-x_i\right)}_{i\in I}=(0{)}_{i\in I}}$ since using the latter as the definition would make the notation useless.

Anet${\displaystyle x_{\bullet }={\left(x_i\right)}_{i\in I}}$ in a TVS${\displaystyle X}$ is called aCauchy net^[4] if$${\displaystyle x_{\bullet }-x_{\bullet }\stackrel{\text{def}}{=}{\left(x_i-x_j\right)}_{(i,j)\in I\times I}\to 0\text{ in }X.}$$ Explicitly, this means that for every neighborhood${\displaystyle N}$ of${\displaystyle 0}$ in${\displaystyle X,}$ there exists some index${\displaystyle i_0\in I}$ such that${\displaystyle x_i-x_j\in N}$ for all indices${\displaystyle i,j\in I}$ that satisfy${\displaystyle i\ge i_0}$ and${\displaystyle j\ge i_0.}$ It suffices to check any of these defining conditions for any givenneighborhood basis of${\displaystyle 0}$ in${\displaystyle X.}$ ACauchy sequence is a sequence that is also a Cauchy net.

If${\displaystyle x_{\bullet }\to x}$ then${\displaystyle x_{\bullet }\times x_{\bullet }\to (x,x)}$ in${\displaystyle X\times X}$ and so the continuity of the vector subtraction map${\displaystyle S:X\times X\to X,}$ which is defined by${\displaystyle S(x,y)\stackrel{\text{def}}{=}x-y,}$ guarantees that${\displaystyle S\left(x_{\bullet }\times x_{\bullet }\right)\to S(x,x)}$ in${\displaystyle X,}$ where${\displaystyle S\left(x_{\bullet }\times x_{\bullet }\right)={\left(x_i-x_j\right)}_{(i,j)\in I\times I}=x_{\bullet }-x_{\bullet }}$ and${\displaystyle S(x,x)=x-x=0.}$This proves that every convergent net is a Cauchy net. By definition, a space is calledcomplete if the converse is also always true. That is,${\displaystyle X}$ is complete if and only if the following holds:

whenever${\displaystyle x_{\bullet }}$ is a net in${\displaystyle X,}$ then${\displaystyle x_{\bullet }}$ converges (to some point) in${\displaystyle X}$ if and only if${\displaystyle x_{\bullet }-x_{\bullet }\to 0}$ in${\displaystyle X.}$

A similar characterization of completeness holds if filters and prefilters are used instead of nets.

A series${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{x}_i}$ is called aCauchy series (respectively, aconvergent series) if the sequence ofpartial sums${\displaystyle {\left(\sum _{i=1}^n{x}_i\right)}_{n=1}^{\mathrm{\infty }}}$ is aCauchy sequence (respectively, aconvergent sequence).^[5] Every convergent series is necessarily a Cauchy series. In a complete TVS, every Cauchy series is necessarily a convergent series.

Aprefilter${\displaystyle \mathcal{B}}$ on atopological vector space${\displaystyle X}$ is called aCauchy prefilter^[6] if it satisfies any of the following equivalent conditions:

1. ${\displaystyle \mathcal{B}-\mathcal{B}\to 0}$ in${\displaystyle X.}$
   • The family${\displaystyle \mathcal{B}-\mathcal{B}\stackrel{\text{def}}{=}\{B-C:B,C\in \mathcal{B}\}}$ is a prefilter.
   • Explicitly,${\displaystyle \mathcal{B}-\mathcal{B}\to 0}$ means that for every neighborhood${\displaystyle N}$ of the origin in${\displaystyle X,}$ there exist${\displaystyle B,C\in \mathcal{B}}$ such that${\displaystyle B-C\subseteq N.}$
2. ${\displaystyle \{B-B:B\in \mathcal{B}\}\to 0}$ in${\displaystyle X.}$
   • The family${\displaystyle \{B-B:B\in \mathcal{B}\}}$ is a prefilter equivalent to${\displaystyle \mathcal{B}-\mathcal{B}}$ (equivalence means these prefilters generate the same filter on${\displaystyle X}$).
   • Explicitly,${\displaystyle \{B-B:B\in \mathcal{B}\}\to 0}$ means that for every neighborhood${\displaystyle N}$ of the origin in${\displaystyle X,}$ there exists some${\displaystyle B\in \mathcal{B}}$ such that${\displaystyle B-B\subseteq N.}$
3. For every neighborhood${\displaystyle N}$ of the origin in${\displaystyle X,}$${\displaystyle \mathcal{B}}$ contains some${\displaystyle N}$-small set (that is, there exists some${\displaystyle B\in \mathcal{B}}$ such that${\displaystyle B-B\subseteq N}$).^[6]
   • A subset${\displaystyle B\subseteq X}$ is called${\displaystyle N}$-small orsmall of order${\displaystyle N}$^[6] if${\displaystyle B-B\subseteq N.}$
4. For every neighborhood${\displaystyle N}$ of the origin in${\displaystyle X,}$ there exists some${\displaystyle x\in X}$ and some${\displaystyle B\in \mathcal{B}}$ such that${\displaystyle B\subseteq x+N.}$^[6]
   • This statement remains true if "${\displaystyle B\subseteq x+N}$" is replaced with "${\displaystyle x+B\subseteq N.}$"
5. Every neighborhood of the origin in${\displaystyle X}$ contains some subset of the form${\displaystyle x+B}$ where${\displaystyle x\in X}$ and${\displaystyle B\in \mathcal{B}.}$

It suffices to check any of the above conditions for any givenneighborhood basis of${\displaystyle 0}$ in${\displaystyle X.}$ ACauchy filter is a Cauchy prefilter that is also afilter on${\displaystyle X.}$

If${\displaystyle \mathcal{B}}$ is a prefilter on a topological vector space${\displaystyle X}$ and if${\displaystyle x\in X,}$ then${\displaystyle \mathcal{B}\to x}$ in${\displaystyle X}$ if and only if${\displaystyle x\in \mathrm{cl}\mathcal{B}}$ and${\displaystyle \mathcal{B}}$ is Cauchy.^[3]

For any${\displaystyle S\subseteq X,}$ a prefilter${\displaystyle \mathcal{C}}$on${\displaystyle S}$ is necessarily a subset of${\displaystyle \mathrm{\wp }(S)}$; that is,${\displaystyle \mathcal{C}\subseteq \mathrm{\wp }(S).}$

A subset${\displaystyle S}$ of a TVS${\displaystyle (X,\tau )}$ is called acomplete subset if it satisfies any of the following equivalent conditions:

1. Every Cauchy prefilter${\displaystyle \mathcal{C}\subseteq \mathrm{\wp }(S)}$ on${\displaystyle S}$converges to at least one point of${\displaystyle S.}$
   • If${\displaystyle X}$ is Hausdorff then every prefilter on${\displaystyle S}$ will converge to at most one point of${\displaystyle X.}$ But if${\displaystyle X}$ is not Hausdorff then a prefilter may converge to multiple points in${\displaystyle X.}$ The same is true for nets.
2. Every Cauchy net in${\displaystyle S}$converges to at least one point of${\displaystyle S.}$
3. ${\displaystyle S}$ is acomplete uniform space (under the point-set topology definition of "complete uniform space") when${\displaystyle S}$ is endowed with the uniformity induced on it by the canonical uniformity of${\displaystyle X.}$

The subset${\displaystyle S}$ is called asequentially complete subset if every Cauchy sequence in${\displaystyle S}$ (or equivalently, every elementary Cauchy filter/prefilter on${\displaystyle S}$) converges to at least one point of${\displaystyle S.}$

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

As a consequence, if a TVS${\displaystyle X}$ isnot Hausdorff then every subset of the closure of${\displaystyle \{0\}}$ in${\displaystyle X}$ is complete because it is compact and every compact set is necessarily complete. In particular, if${\displaystyle \varnothing \ne S\subseteq {\mathrm{cl}}_X\{0\}}$ is a proper subset, such as${\displaystyle S=\{0\}}$ for example, then${\displaystyle S}$ would be complete even thoughevery Cauchy net in${\displaystyle S}$ (and also every Cauchy prefilter on${\displaystyle S}$) converges toevery point in${\displaystyle {\mathrm{cl}}_X\{0\},}$ including those points in${\displaystyle {\mathrm{cl}}_X\{0\}}$ that do not belong to${\displaystyle S.}$ 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${\displaystyle \varnothing \ne S\subseteq {\mathrm{cl}}_X\{0\}}$ then${\displaystyle S={\mathrm{cl}}_X\{0\}}$ if and only if${\displaystyle S}$ is closed in${\displaystyle X.}$

Atopological vector space${\displaystyle X}$ is called acomplete topological vector space if any of the following equivalent conditions are satisfied:

1. ${\displaystyle X}$ is acomplete uniform space when it is endowed with its canonical uniformity.
   • In the general theory ofuniform spaces, a uniform space is called acomplete uniform space if eachCauchy filter on${\displaystyle X}$ converges to some point of${\displaystyle X}$ in the topology induced by the uniformity. When${\displaystyle X}$ is a TVS, the topology induced by the canonical uniformity is equal to${\displaystyle X}$'s given topology (so convergence in this induced topology is just the usual convergence in${\displaystyle X}$).
2. ${\displaystyle X}$ is a complete subset of itself.
3. There exists a neighborhood of the origin in${\displaystyle X}$ that is also a complete subset of${\displaystyle X.}$^[6]
   • This implies that everylocally compact TVS is complete (even if the TVS is not Hausdorff).
4. Every Cauchy prefilter${\displaystyle \mathcal{C}\subseteq \mathrm{\wp }(X)}$ on${\displaystyle X}$converges in${\displaystyle X}$ to at least one point of${\displaystyle X.}$
   • If${\displaystyle X}$ is Hausdorff then every prefilter on${\displaystyle X}$ will converge to at most one point of${\displaystyle X.}$ But if${\displaystyle X}$ is not Hausdorff then a prefilter may converge to multiple points in${\displaystyle X.}$ The same is true for nets.
5. Every Cauchyfilter on${\displaystyle X}$ converges in${\displaystyle X}$ to at least one point of${\displaystyle X.}$
6. Every Cauchy net in${\displaystyle X}$converges in${\displaystyle X}$ to at least one point of${\displaystyle X.}$

where if in addition${\displaystyle X}$ ispseudometrizable or metrizable (for example, anormed space) then this list can be extended to include:

7. ${\displaystyle X}$ is sequentially complete.

A topological vector space${\displaystyle X}$ issequentially complete if any of the following equivalent conditions are satisfied:

1. ${\displaystyle X}$ is a sequentially complete subset of itself.
2. Every Cauchy sequence in${\displaystyle X}$ converges in${\displaystyle X}$ to at least one point of${\displaystyle X.}$
3. Every elementary Cauchy prefilter on${\displaystyle X}$ converges in${\displaystyle X}$ to at least one point of${\displaystyle X.}$
4. Every elementary Cauchy filter on${\displaystyle X}$ converges in${\displaystyle X}$ to at least one point of${\displaystyle X.}$

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

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

For any subsets${\displaystyle \mathrm{\Phi },\mathrm{\Psi }\subseteq X\times X,}$ let^[1]$${\displaystyle {\mathrm{\Phi }}^{\mathrm{op}}\stackrel{\text{def}}{=}\{(y,x):(x,y)\in \mathrm{\Phi }\}}$$and letA non-empty family${\displaystyle \mathcal{B}\subseteq \mathrm{\wp }(X\times X)}$ is called abase of entourages or afundamental system of entourages if${\displaystyle \mathcal{B}}$ is aprefilter on${\displaystyle X\times X}$ satisfying all of the following conditions:

1. Every set in${\displaystyle \mathcal{B}}$ contains the diagonal of${\displaystyle X}$ as a subset; that is,${\displaystyle {\mathrm{\Delta }}_X\stackrel{\text{def}}{=}\{(x,x):x\in X\}\subseteq \mathrm{\Phi }}$ for every${\displaystyle \mathrm{\Phi }\in \mathcal{B}.}$ Said differently, the prefilter${\displaystyle \mathcal{B}}$ isfixed on${\displaystyle {\mathrm{\Delta }}_X.}$
2. For every${\displaystyle \mathrm{\Omega }\in \mathcal{B}}$ there exists some${\displaystyle \mathrm{\Phi }\in \mathcal{B}}$ such that${\displaystyle \mathrm{\Phi }\circ \mathrm{\Phi }\subseteq \mathrm{\Omega }.}$
3. For every${\displaystyle \mathrm{\Omega }\in \mathcal{B}}$ there exists some${\displaystyle \mathrm{\Phi }\in \mathcal{B}}$ such that${\displaystyle \mathrm{\Phi }\subseteq {\mathrm{\Omega }}^{\mathrm{op}}\stackrel{\text{def}}{=}\{(y,x):(x,y)\in \mathrm{\Omega }\}.}$

Auniformity oruniform structure on${\displaystyle X}$ is afilter${\displaystyle \mathcal{U}}$ on${\displaystyle X\times X}$ that is generated by some base of entourages${\displaystyle \mathcal{B},}$ in which case we say that${\displaystyle \mathcal{B}}$ is abase of entouragesfor${\displaystyle \mathcal{U}.}$

For a commutative additive group${\displaystyle X,}$ atranslation-invariant fundamental system of entourages^[7] is a fundamental system of entourages${\displaystyle \mathcal{B}}$ such that for every${\displaystyle \mathrm{\Phi }\in \mathcal{B},}$${\displaystyle (x,y)\in \mathrm{\Phi }}$ if and only if${\displaystyle (x+z,y+z)\in \mathrm{\Phi }}$ for all${\displaystyle x,y,z\in X.}$ A uniformity${\displaystyle \mathcal{B}}$ is called atranslation-invariant uniformity^[7] if it has a base of entourages that is translation-invariant. The canonical uniformity on any TVS is translation-invariant.^[7]

The binary operator${\displaystyle \circ }$ satisfies all of the following:

• ${\displaystyle (\mathrm{\Phi }\circ \mathrm{\Psi }{)}^{\mathrm{op}}={\mathrm{\Psi }}^{\mathrm{op}}\circ {\mathrm{\Phi }}^{\mathrm{op}}.}$
• If${\displaystyle \mathrm{\Phi }\subseteq {\mathrm{\Phi }}_2}$ and${\displaystyle \mathrm{\Psi }\subseteq {\mathrm{\Psi }}_2}$ then${\displaystyle \mathrm{\Phi }\circ \mathrm{\Psi }\subseteq {\mathrm{\Phi }}_2\circ {\mathrm{\Psi }}_2.}$
• Associativity:${\displaystyle \mathrm{\Phi }\circ (\mathrm{\Psi }\circ \mathrm{\Omega })=(\mathrm{\Phi }\circ \mathrm{\Psi })\circ \mathrm{\Omega }.}$
• Identity:${\displaystyle \mathrm{\Phi }\circ {\mathrm{\Delta }}_X=\mathrm{\Phi }={\mathrm{\Delta }}_X\circ \mathrm{\Phi }.}$
• Zero:${\displaystyle \mathrm{\Phi }\circ \varnothing =\varnothing =\varnothing \circ \mathrm{\Phi }}$

Symmetric entourages

Call a subset${\displaystyle \mathrm{\Phi }\subseteq X\times X}$symmetric if${\displaystyle \mathrm{\Phi }={\mathrm{\Phi }}^{\mathrm{op}},}$ which is equivalent to${\displaystyle {\mathrm{\Phi }}^{\mathrm{op}}\subseteq \mathrm{\Phi }.}$ This equivalence follows from the identity${\displaystyle {\left({\mathrm{\Phi }}^{\mathrm{op}}\right)}^{\mathrm{op}}=\mathrm{\Phi }}$ and the fact that if${\displaystyle \mathrm{\Psi }\subseteq X\times X,}$ then${\displaystyle \mathrm{\Phi }\subseteq \mathrm{\Psi }}$ if and only if${\displaystyle {\mathrm{\Phi }}^{\mathrm{op}}\subseteq {\mathrm{\Psi }}^{\mathrm{op}}.}$ For example, the set${\displaystyle {\mathrm{\Phi }}^{\mathrm{op}}\cap \mathrm{\Phi }}$ is always symmetric for every${\displaystyle \mathrm{\Phi }\subseteq X\times X.}$ And because${\displaystyle (\mathrm{\Phi }\cap \mathrm{\Psi }{)}^{\mathrm{op}}={\mathrm{\Phi }}^{\mathrm{op}}\cap {\mathrm{\Psi }}^{\mathrm{op}},}$ if${\displaystyle \mathrm{\Phi }}$ and${\displaystyle \mathrm{\Psi }}$ are symmetric then so is${\displaystyle \mathrm{\Phi }\cap \mathrm{\Psi }.}$

Relatives

Let${\displaystyle \mathrm{\Phi }\subseteq X\times X}$ be arbitrary and let${\displaystyle {\mathrm{Pr}}_1,{\mathrm{Pr}}_2:X\times X\to X}$ be the canonical projections onto the first and second coordinates, respectively.

For any${\displaystyle S\subseteq X,}$ define$${\displaystyle S\cdot \mathrm{\Phi }\stackrel{\text{def}}{=}\{y\in X:\mathrm{\Phi }\cap (S\times \{x\})\ne \varnothing \}={\mathrm{Pr}}_2(\mathrm{\Phi }\cap (S\times X))}$$$${\displaystyle \mathrm{\Phi }\cdot S\stackrel{\text{def}}{=}\{x\in X:\mathrm{\Phi }\cap (\{x\}\times S)\ne \varnothing \}={\mathrm{Pr}}_1(\mathrm{\Phi }\cap (X\times S))=S\cdot \left({\mathrm{\Phi }}^{\mathrm{op}}\right)}$$where${\displaystyle \mathrm{\Phi }\cdot S}$ (respectively,${\displaystyle S\cdot \mathrm{\Phi }}$) is called the set ofleft (respectively,right)${\displaystyle \mathrm{\Phi }}$-relatives of (points in)${\displaystyle S.}$ Denote the special case where${\displaystyle S=\{p\}}$ is a singleton set for some${\displaystyle p\in X}$ by:$${\displaystyle p\cdot \mathrm{\Phi }\stackrel{\text{def}}{=}\{p\}\cdot \mathrm{\Phi }=\{y\in X:(p,y)\in \mathrm{\Phi }\}}$$$${\displaystyle \mathrm{\Phi }\cdot p\stackrel{\text{def}}{=}\mathrm{\Phi }\cdot \{p\}=\{x\in X:(x,p)\in \mathrm{\Phi }\}=p\cdot \left({\mathrm{\Phi }}^{\mathrm{op}}\right)}$$If${\displaystyle \mathrm{\Phi },\mathrm{\Psi }\subseteq X\times X}$ then$(\mathrm{\Phi }\circ \mathrm{\Psi })\cdot S=\mathrm{\Phi }\cdot (\mathrm{\Psi }\cdot S).$ Moreover,${\displaystyle \cdot }$right distributes over both unions and intersections, meaning that if${\displaystyle R,S\subseteq X}$ then${\displaystyle (R\cup S)\cdot \mathrm{\Phi }=(R\cdot \mathrm{\Phi })\cup (S\cdot \mathrm{\Phi })}$ and${\displaystyle (R\cap S)\cdot \mathrm{\Phi }\subseteq (R\cdot \mathrm{\Phi })\cap (S\cdot \mathrm{\Phi }).}$

Neighborhoods and open sets

Two points${\displaystyle x}$ and${\displaystyle y}$ are${\displaystyle \mathrm{\Phi }}$-close if${\displaystyle (x,y)\in \mathrm{\Phi }}$ and a subset${\displaystyle S\subseteq X}$ is called${\displaystyle \mathrm{\Phi }}$-small if${\displaystyle S\times S\subseteq \mathrm{\Phi }.}$

Let${\displaystyle \mathcal{B}\subseteq \mathrm{\wp }(X\times X)}$ be a base of entourages on${\displaystyle X.}$ Theneighborhood prefilter at a point${\displaystyle p\in X}$ and, respectively, on a subset${\displaystyle S\subseteq X}$ are thefamilies of sets:$${\displaystyle \mathcal{B}\cdot p\stackrel{\text{def}}{=}\mathcal{B}\cdot \{p\}=\{\mathrm{\Phi }\cdot p:\mathrm{\Phi }\in \mathcal{B}\}\text{ and }\mathcal{B}\cdot S\stackrel{\text{def}}{=}\{\mathrm{\Phi }\cdot S:\mathrm{\Phi }\in \mathcal{B}\}}$$and the filters on${\displaystyle X}$ that each generates is known as theneighborhood filter of${\displaystyle p}$ (respectively, of${\displaystyle S}$). Assign to every${\displaystyle x\in X}$ the neighborhood prefilter$${\displaystyle \mathcal{B}\cdot x\stackrel{\text{def}}{=}\{\mathrm{\Phi }\cdot x:\mathrm{\Phi }\in \mathcal{B}\}}$$and use theneighborhood definition of "open set" to obtain atopology on${\displaystyle X}$ called thetopology induced by${\displaystyle \mathcal{B}}$ or theinduced topology. Explicitly, a subset${\displaystyle U\subseteq X}$ is open in this topology if and only if for every${\displaystyle u\in U}$ there exists some${\displaystyle N\in \mathcal{B}\cdot u}$ such that${\displaystyle N\subseteq U;}$ that is,${\displaystyle U}$ is open if and only if for every${\displaystyle u\in U}$ there exists some${\displaystyle \mathrm{\Phi }\in \mathcal{B}}$ such that${\displaystyle \mathrm{\Phi }\cdot u\stackrel{\text{def}}{=}\{x\in X:(x,u)\in \mathrm{\Phi }\}\subseteq U.}$

The closure of a subset${\displaystyle S\subseteq X}$ in this topology is:$${\displaystyle {\mathrm{cl}}_{X}S=\bigcap _{\mathrm{\Phi }\in \mathcal{B}}(\mathrm{\Phi }\cdot S)=\bigcap _{\mathrm{\Phi }\in \mathcal{B}}(S\cdot \mathrm{\Phi }).}$$

Cauchy prefilters and complete uniformities

A prefilter${\displaystyle \mathcal{F}\subseteq \mathrm{\wp }(X)}$ on a uniform space${\displaystyle X}$ with uniformity${\displaystyle \mathcal{U}}$ is called aCauchy prefilter if for every entourage${\displaystyle N\in \mathcal{U},}$ there exists some${\displaystyle F\in \mathcal{F}}$ such that${\displaystyle F\times F\subseteq N.}$

A uniform space${\displaystyle (X,\mathcal{U})}$ is called acomplete uniform space (respectively, asequentially complete uniform space) if every Cauchy prefilter (respectively, every elementary Cauchy prefilter) on${\displaystyle X}$ converges to at least one point of${\displaystyle X}$ when${\displaystyle X}$ is endowed with the topology induced by${\displaystyle \mathcal{U}.}$

Case of a topological vector space

If${\displaystyle (X,\tau )}$ is atopological vector space then for any${\displaystyle S\subseteq X}$ and${\displaystyle x\in X,}$$${\displaystyle {\mathrm{\Delta }}_X(N)\cdot S=S+N\text{ and }{\mathrm{\Delta }}_X(N)\cdot x=x+N,}$$and the topology induced on${\displaystyle X}$ by the canonical uniformity is the same as the topology that${\displaystyle X}$ started with (that is, it is${\displaystyle \tau }$).

Let${\displaystyle X}$ and${\displaystyle Y}$ be TVSs,${\displaystyle D\subseteq X,}$ and${\displaystyle f:D\to Y}$ be a map. Then${\displaystyle f:D\to Y}$ isuniformly continuous if for every neighborhood${\displaystyle U}$ of the origin in${\displaystyle X,}$ there exists a neighborhood${\displaystyle V}$ of the origin in${\displaystyle Y}$ such that for all${\displaystyle x,y\in D,}$ if${\displaystyle y-x\in U}$ then${\displaystyle f(y)-f(x)\in V.}$

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

We review the basic notions related to the general theory of complete pseudometric spaces. Recall that everymetric is apseudometric and that a pseudometric${\displaystyle p}$ is a metric if and only if${\displaystyle p(x,y)=0}$ implies${\displaystyle x=y.}$ Thus everymetric space is apseudometric space and a pseudometric space${\displaystyle (X,p)}$ is a metric space if and only if${\displaystyle p}$ is a metric.

If${\displaystyle S}$ is a subset of apseudometric space${\displaystyle (X,d)}$ then thediameter of${\displaystyle S}$ is defined to be$${\displaystyle \mathrm{diam}(S)\stackrel{\text{def}}{=}\underset{}{sup}\{d(s,t):s,t\in S\}.}$$

A prefilter${\displaystyle \mathcal{B}}$ on a pseudometric space${\displaystyle (X,d)}$ is called a${\displaystyle d}$-Cauchy prefilter or simply aCauchy prefilter if for eachreal${\displaystyle r>0,}$ there is some${\displaystyle B\in \mathcal{B}}$ such that the diameter of${\displaystyle B}$ is less than${\displaystyle r.}$

Suppose${\displaystyle (X,d)}$ is a pseudometric space. Anet${\displaystyle x_{\bullet }={\left(x_i\right)}_{i\in I}}$ in${\displaystyle X}$ is called a${\displaystyle d}$-Cauchy net or simply aCauchy net if${\displaystyle \mathrm{Tails}\left(x_{\bullet }\right)}$ is a Cauchy prefilter, which happens if and only if

for every${\displaystyle r>0}$ there is some${\displaystyle i\in I}$ such that if${\displaystyle j,k\in I}$ with${\displaystyle j\ge i}$ and${\displaystyle k\ge i}$ then

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

ACauchy sequence is a sequence that is also a Cauchy net.^{[note 3]}

Every pseudometric${\displaystyle p}$ on a set${\displaystyle X}$ induces the usual canonical topology on${\displaystyle X,}$ which we'll denote by${\displaystyle {\tau }_p}$; it also induces a canonicaluniformity on${\displaystyle X,}$ which we'll denote by${\displaystyle {\mathcal{U}}_p.}$ The topology on${\displaystyle X}$ induced by the uniformity${\displaystyle {\mathcal{U}}_p}$ is equal to${\displaystyle {\tau }_p.}$ A net${\displaystyle x_{\bullet }={\left(x_i\right)}_{i\in I}}$ in${\displaystyle X}$ is Cauchy with respect to${\displaystyle p}$ if and only if it is Cauchy with respect to the uniformity${\displaystyle {\mathcal{U}}_p.}$ The pseudometric space${\displaystyle (X,p)}$ is acomplete (resp. a sequentially complete) pseudometric space if and only if${\displaystyle \left(X,{\mathcal{U}}_p\right)}$ is acomplete (resp. a sequentially complete) uniform space. Moreover, the pseudometric space${\displaystyle (X,p)}$ (resp. the uniform space${\displaystyle \left(X,{\mathcal{U}}_p\right)}$) is complete if and only if it is sequentially complete.

A pseudometric space${\displaystyle (X,d)}$ (for example, ametric space) is calledcomplete and${\displaystyle d}$ is called acomplete pseudometric if any of the following equivalent conditions hold:

1. Every Cauchy prefilter on${\displaystyle X}$ converges to at least one point of${\displaystyle X.}$
2. The above statement but with the word "prefilter" replaced by "filter."
3. Every Cauchy net in${\displaystyle X}$ converges to at least one point of${\displaystyle X.}$
   • If${\displaystyle d}$ is a metric on${\displaystyle X}$ then any limit point is necessarily unique and the same is true for limits of Cauchy prefilters on${\displaystyle X.}$
4. Every Cauchy sequence in${\displaystyle X}$ converges to at least one point of${\displaystyle X.}$
   • Thus to prove that${\displaystyle (X,d)}$ is complete, it suffices to only consider Cauchy sequences in${\displaystyle X}$ (and it is not necessary to consider the more general Cauchy nets).
5. The canonical uniformity on${\displaystyle X}$ induced by the pseudometric${\displaystyle d}$ is a complete uniformity.

And if addition${\displaystyle d}$ is a metric then we may add to this list:

6. Every decreasing sequence of closed balls whose diameters shrink to${\displaystyle 0}$ has non-empty intersection.^[8]

EveryF-space, and thus also everyFréchet space,Banach space, andHilbert space is a complete TVS. Note that everyF-space is aBaire space but there are normed spaces that are Baire but not Banach.^[9]

A pseudometric${\displaystyle d}$ on a vector space${\displaystyle X}$ is said to be atranslation invariant pseudometric if${\displaystyle d(x,y)=d(x+z,y+z)}$ for all vectors${\displaystyle x,y,z\in X.}$

Suppose${\displaystyle (X,\tau )}$ ispseudometrizable TVS (for example, a metrizable TVS) and that${\displaystyle p}$ isany pseudometric on${\displaystyle X}$ such that the topology on${\displaystyle X}$ induced by${\displaystyle p}$ is equal to${\displaystyle \tau .}$ If${\displaystyle p}$ is translation-invariant, then${\displaystyle (X,\tau )}$ is a complete TVS if and only if${\displaystyle (X,p)}$ is a complete pseudometric space.^[10] If${\displaystyle p}$ isnot translation-invariant, then may be possible for${\displaystyle (X,\tau )}$ to be a complete TVS but${\displaystyle (X,p)}$ tonot be a complete pseudometric space^[10] (see this footnote^{[note 4]} for an example).^[10]