Complete topological vector space

In functional analysis and related areas of mathematics, a complete topological vector space is a topological 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 by Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, while "point ${\displaystyle x}$ towards which they all get closer" means that this Cauchy net or filter converges to ${\displaystyle x.}$ The notion of completeness for TVSs uses the theory of uniform spaces as a framework to generalize the notion of completeness for metric spaces. But unlike metric-completeness, TVS-completeness does not depend on any metric and is defined for all TVSs, including those that are not metrizable or Hausdorff.

Completeness is an extremely important property for a topological vector space to possess. The notions of completeness for normed spaces and metrizable TVSs, 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 – a notion that is independent of any particular norm or metric. A metrizable topological vector space ${\displaystyle X}$ with a translation invariant metric^{[note 1]} ${\displaystyle d}$ is complete as a TVS if and only if ${\displaystyle (X,d)}$ is a complete 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 also metrizable include all F-spaces and consequently also all Fréchet spaces, Banach spaces, and Hilbert spaces. Prominent examples of complete TVS that are (typically) not metrizable include strict LF-spaces such as the space of test functions ${\displaystyle C_{c}^{\infty }(U)}$ with it canonical LF-topology, the strong dual space of any non-normable Fréchet space, as well as many other polar topologies on continuous dual space or other topologies on spaces of linear maps.

Explicitly, a topological vector spaces (TVS) is complete if every net, or equivalently, every filter, that is Cauchy with respect to the space's 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 ${\displaystyle (X,\tau )}$ is the unique^{[note 2]} translation-invariant uniformity that induces on ${\displaystyle X}$ the topology ${\displaystyle \tau .}$ This notion of "TVS-completeness" depends 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 metrics or 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 ${\displaystyle X,}$ even if it is not metrizable or not Hausdorff, has a completion, which by definition is a complete TVS ${\displaystyle C}$ into which ${\displaystyle X}$ can be TVS-embedded as a dense vector subspace. Moreover, every Hausdorff TVS has a Hausdorff completion, which is necessarily unique up to TVS-isomorphism. However, as discussed below, all TVSs have infinitely many non-Hausdorff completions that are not TVS-isomorphic to one another.

Definitions[edit]

This section summarizes the definition of a complete topological vector space (TVS) in terms of both 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.

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

Canonical uniformity[edit]

The diagonal of ${\displaystyle X}$ is the set^[1]

and for any ${\displaystyle N\subseteq X,}$ the canonical entourage/vicinity around ${\displaystyle N}$ is the set where if ${\displaystyle 0\in N}$ then ${\displaystyle \Delta _{X}(N)}$ contains the diagonal ${\displaystyle \Delta _{X}(\{0\})=\Delta _{X}.}$

If ${\displaystyle N}$ is a symmetric set (that is, if ${\displaystyle -N=N}$), then ${\displaystyle \Delta _{X}(N)}$ is symmetric, which by definition means that ${\displaystyle \Delta _{X}(N)=\left(\Delta _{X}(N)\right)^{\operatorname {op} }}$ holds where ${\displaystyle \left(\Delta _{X}(N)\right)^{\operatorname {op} }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{(y,x):(x,y)\in \Delta _{X}(N)\right\},}$ and in addition, this symmetric set's composition with itself is:

If ${\displaystyle {\mathcal {L}}}$ is any neighborhood basis at the origin in ${\displaystyle (X,\tau )}$ then the family of subsets of ${\displaystyle X\times X:}$

is a prefilter on ${\displaystyle X\times X.}$ If ${\displaystyle {\mathcal {N}}_{\tau }(0)}$ is the neighborhood filter at the origin in ${\displaystyle (X,\tau )}$ then ${\displaystyle {\mathcal {B}}_{{\mathcal {N}}_{\tau }(0)}}$ forms a base of entourages for a uniform structure on ${\displaystyle X}$ that is considered canonical.^[2] Explicitly, by definition, the canonical uniformity on ${\displaystyle X}$ induced by ${\displaystyle (X,\tau )}$^[2] is the filter ${\displaystyle {\mathcal {U}}_{\tau }}$ on ${\displaystyle X\times X}$ generated by the above prefilter: where ${\displaystyle {\mathcal {B}}_{{\mathcal {N}}_{\tau }(0)}^{\uparrow }}$ denotes the upward 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 ${\displaystyle (X,\tau ).}$

Cauchy net[edit]

The general theory of uniform 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 a directed set by declaring ${\displaystyle (i,j)\leq \left(i_{2},j_{2}\right)}$ if and only if ${\displaystyle i\leq i_{2}}$ and ${\displaystyle j\leq j_{2}.}$ Then

denotes the (Cartesian) product net, where in particular ${\textstyle x_{\bullet }\times x_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(x_{i},x_{j}\right)_{(i,j)\in I\times I}.}$ If ${\displaystyle X=Y}$ then the image of this net under the vector addition map ${\displaystyle X\times X\to X}$ denotes the sum of these two nets:^[3] and similarly their difference is defined to be the image of the product net under the vector subtraction map ${\displaystyle (x,y)\mapsto x-y}$: 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.

A net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ in a TVS ${\displaystyle X}$ is called a Cauchy net^[4] if

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 ${\displaystyle i\geq i_{0}}$ and ${\displaystyle j\geq i_{0}.}$ It suffices to check any of these defining conditions for any given neighborhood basis of ${\displaystyle 0}$ in ${\displaystyle X.}$ A Cauchy 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 ${\displaystyle S(x,y)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~x-y,}$ 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 called complete 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 ${\displaystyle \sum _{i=1}^{\infty }x_{i}}$ is called a Cauchy series (respectively, a convergent series) if the sequence of partial sums ${\displaystyle \left(\sum _{i=1}^{n}x_{i}\right)_{n=1}^{\infty }}$ is a Cauchy sequence (respectively, a convergent sequence).^[5] 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[edit]

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

1. ${\displaystyle {\mathcal {B}}-{\mathcal {B}}\to 0}$ in ${\displaystyle X.}$
   • The family ${\displaystyle {\mathcal {B}}-{\mathcal {B}}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{B-C:B,C\in {\mathcal {B}}\}}$ 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 or small 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 given neighborhood basis of ${\displaystyle 0}$ in ${\displaystyle X.}$ A Cauchy filter is a Cauchy prefilter that is also a filter 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 ${\displaystyle x\in \operatorname {cl} {\mathcal {B}}}$ and ${\displaystyle {\mathcal {B}}}$ is Cauchy.^[3]

Complete subset[edit]

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

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

1. Every Cauchy prefilter ${\displaystyle {\mathcal {C}}\subseteq \wp (S)}$ 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 a complete 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 a sequentially 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 ${\displaystyle S}$ converge only 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 ${\displaystyle X}$ is not 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 ${\displaystyle \varnothing \neq S\subseteq \operatorname {cl} _{X}\{0\}}$ is a proper subset, such as ${\displaystyle S=\{0\}}$ for example, then ${\displaystyle S}$ would be complete even though every Cauchy net in ${\displaystyle S}$ (and also every Cauchy prefilter on ${\displaystyle S}$) converges to every point in ${\displaystyle \operatorname {cl} _{X}\{0\},}$ including those points in ${\displaystyle \operatorname {cl} _{X}\{0\}}$ 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 ${\displaystyle \varnothing \neq S\subseteq \operatorname {cl} _{X}\{0\}}$ then ${\displaystyle S=\operatorname {cl} _{X}\{0\}}$ if and only if ${\displaystyle S}$ is closed in ${\displaystyle X.}$

Complete topological vector space[edit]

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

1. ${\displaystyle X}$ is a complete uniform space when it is endowed with its canonical uniformity.
   • In the general theory of uniform spaces, a uniform space is called a complete uniform space if each Cauchy 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 every locally compact TVS is complete (even if the TVS is not Hausdorff).
4. Every Cauchy prefilter ${\displaystyle {\mathcal {C}}\subseteq \wp (X)}$ 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 Cauchy filter 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 ${\displaystyle X}$ is pseudometrizable or metrizable (for example, a normed space) then this list can be extended to include:

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

A topological vector space ${\displaystyle X}$ is sequentially 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 ${\displaystyle X.}$

Uniqueness of the canonical uniformity[edit]

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.

Uniform spaces and translation-invariant uniformities[edit]

For any subsets ${\displaystyle \Phi ,\Psi \subseteq X\times X,}$ let^[1]

and let A non-empty family ${\displaystyle {\mathcal {B}}\subseteq \wp (X\times X)}$ is called a base of entourages or a fundamental system of entourages if ${\displaystyle {\mathcal {B}}}$ is a prefilter 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, ${\displaystyle \Delta _{X}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{(x,x):x\in X\}\subseteq \Phi }$ for every ${\displaystyle \Phi \in {\mathcal {B}}.}$ Said differently, the prefilter ${\displaystyle {\mathcal {B}}}$ is fixed on ${\displaystyle \Delta _{X}.}$
2. For every ${\displaystyle \Omega \in {\mathcal {B}}}$ there exists some ${\displaystyle \Phi \in {\mathcal {B}}}$ such that ${\displaystyle \Phi \circ \Phi \subseteq \Omega .}$
3. For every ${\displaystyle \Omega \in {\mathcal {B}}}$ there exists some ${\displaystyle \Phi \in {\mathcal {B}}}$ such that ${\displaystyle \Phi \subseteq \Omega ^{\operatorname {op} }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{(y,x):(x,y)\in \Omega \}.}$

A uniformity or uniform structure on ${\displaystyle X}$ is a filter ${\displaystyle {\mathcal {U}}}$ 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 a base of entourages for ${\displaystyle {\mathcal {U}}.}$

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

• ${\displaystyle (\Phi \circ \Psi )^{\operatorname {op} }=\Psi ^{\operatorname {op} }\circ \Phi ^{\operatorname {op} }.}$
• If ${\displaystyle \Phi \subseteq \Phi _{2}}$ and ${\displaystyle \Psi \subseteq \Psi _{2}}$ then ${\displaystyle \Phi \circ \Psi \subseteq \Phi _{2}\circ \Psi _{2}.}$
• Associativity: ${\displaystyle \Phi \circ (\Psi \circ \Omega )=(\Phi \circ \Psi )\circ \Omega .}$
• Identity: ${\displaystyle \Phi \circ \Delta _{X}=\Phi =\Delta _{X}\circ \Phi .}$
• Zero: ${\displaystyle \Phi \circ \varnothing =\varnothing =\varnothing \circ \Phi }$

Symmetric entourages

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

Topology generated by a uniformity[edit]

Relatives

Let ${\displaystyle \Phi \subseteq X\times X}$ be arbitrary and let ${\displaystyle \operatorname {Pr} _{1},\operatorname {Pr} _{2}:X\times X\to X}$ be the canonical projections onto the first and second coordinates, respectively.

For any ${\displaystyle S\subseteq X,}$ define

where ${\displaystyle \Phi \cdot S}$ (respectively, ${\displaystyle S\cdot \Phi }$) is called the set of left (respectively, right) ${\displaystyle \Phi }$-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: If ${\displaystyle \Phi ,\Psi \subseteq X\times X}$ then ${\textstyle (\Phi \circ \Psi )\cdot S=\Phi \cdot (\Psi \cdot S).}$ Moreover, ${\displaystyle \,\cdot \,}$ right distributes over both unions and intersections, meaning that if ${\displaystyle R,S\subseteq X}$ then ${\displaystyle (R\cup S)\cdot \Phi ~=~(R\cdot \Phi )\cup (S\cdot \Phi )}$ and ${\displaystyle (R\cap S)\cdot \Phi ~\subseteq ~(R\cdot \Phi )\cap (S\cdot \Phi ).}$

Neighborhoods and open sets

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

Let ${\displaystyle {\mathcal {B}}\subseteq \wp (X\times X)}$ be a base of entourages on ${\displaystyle X.}$ The neighborhood prefilter at a point ${\displaystyle p\in X}$ and, respectively, on a subset ${\displaystyle S\subseteq X}$ are the families of sets:

and the filters on ${\displaystyle X}$ that each generates is known as the neighborhood filter of ${\displaystyle p}$ (respectively, of ${\displaystyle S}$). Assign to every ${\displaystyle x\in X}$ the neighborhood prefilter and use the neighborhood definition of "open set" to obtain a topology on ${\displaystyle X}$ called the topology induced by ${\displaystyle {\mathcal {B}}}$ or the induced 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 ${\displaystyle \Phi \in {\mathcal {B}}}$ such that ${\displaystyle \Phi \cdot u~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{x\in X:(x,u)\in \Phi \}\subseteq U.}$

The closure of a subset ${\displaystyle S\subseteq X}$ in this topology is:

Cauchy prefilters and complete uniformities

A prefilter ${\displaystyle {\mathcal {F}}\subseteq \wp (X)}$ on a uniform space ${\displaystyle X}$ with uniformity ${\displaystyle {\mathcal {U}}}$ is called a Cauchy 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 a complete uniform space (respectively, a sequentially 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 a topological vector space then for any ${\displaystyle S\subseteq X}$ and ${\displaystyle x\in X,}$

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 ${\displaystyle \tau }$).

Uniform continuity[edit]

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be TVSs, ${\displaystyle D\subseteq X,}$ and ${\displaystyle f:D\to Y}$ be a map. Then ${\displaystyle f:D\to Y}$ is uniformly 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 Cauchy prefilter, it will be a Cauchy filter in ${\displaystyle Y}$ if and only if ${\displaystyle f:D\to Y}$ is surjective.

TVS completeness vs completeness of (pseudo)metrics[edit]

Preliminaries: Complete pseudometric spaces[edit]

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

A prefilter ${\displaystyle {\mathcal {B}}}$ on a pseudometric space ${\displaystyle (X,d)}$ is called a ${\displaystyle d}$-Cauchy prefilter or simply a Cauchy prefilter if for each real ${\displaystyle r>0,}$ 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. A net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ in ${\displaystyle X}$ is called a ${\displaystyle d}$-Cauchy net or simply a Cauchy net if ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)}$ 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 ${\displaystyle j\geq i}$ and ${\displaystyle k\geq i}$ then ${\displaystyle d\left(x_{j},x_{k}\right)<r}$

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 ${\displaystyle \mathbb {R} .}$

A Cauchy 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 canonical uniformity 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 a complete (resp. a sequentially complete) pseudometric space if and only if ${\displaystyle \left(X,{\mathcal {U}}_{p}\right)}$ is a complete (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, a metric space) is called complete and ${\displaystyle d}$ is called a complete 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]

Complete pseudometrics and complete TVSs[edit]

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.^[9]

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

Suppose ${\displaystyle (X,\tau )}$ is pseudometrizable TVS (for example, a metrizable TVS) and that ${\displaystyle p}$ is any 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 ${\displaystyle p}$ is not translation-invariant, then may be possible for ${\displaystyle (X,\tau )}$ to be a complete TVS but ${\displaystyle (X,p)}$ to not be a complete pseudometric space^[10] (see this footnote^{[note 4]} for an example).^[10]

Complete norms and equivalent norms[edit]

Two norms on a vector space are called equivalent if and only if they induce the same topology.^[13] If ${\displaystyle p}$ and ${\displaystyle q}$ are two equivalent norms on a vector space ${\displaystyle X}$ then the normed space ${\displaystyle (X,p)}$ is a Banach space if and only if ${\displaystyle (X,q)}$ is a Banach space. See this footnote for an example of a continuous norm on a Banach space that is not equivalent to that Banach space's given norm.^{[note 6][13]} All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.^[14] 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[edit]

A completion^[15] of a TVS ${\displaystyle X}$ is a complete TVS that contains a dense vector subspace that is TVS-isomorphic to ${\displaystyle X.}$ In other words, it is a complete TVS ${\displaystyle C}$ into which ${\displaystyle X}$ can be TVS-embedded as a dense vector subspace. Every TVS-embedding is a uniform embedding.

Every topological vector space has a completion. Moreover, every Hausdorff TVS has a Hausdorff completion, which is necessarily unique 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 not TVS-isomorphic to one another.

Examples of completions[edit]

For example, the vector spa