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\begin{document}

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\section*{complete topological vector space}

\hypertarget{idea}{}\section*{{Idea}}\label{idea}

As [[TVS|topological vector spaces]] are [[uniform spaces]], it is appropriate to discuss [[complete uniform space|completeness]]. As with a uniform space, a topological vector space is \emph{complete} if it has no holes: everything that should be there actually is there.

Where this gets interesting is in the question as to what \emph{should} be there. To determine this, one has to have some method of discovering holes. This is usually done by means of [[Cauchy nets]], but in a given application it may not be necessary that all holes be filled and this leads to weaker notions of completeness.

The more general notions make sense for arbitrary [[topological vector spaces]] but some more refined notions are only used for [[locally convex topological vector spaces]].

\hypertarget{definitions}{}\section*{{Definitions}}\label{definitions}

In strict order of decreasing strength, we have the following notions of completeness.

\begin{defn}
\label{WeakCplt}\hypertarget{WeakCplt}{}
A [[locally convex topological vector space]] is \textbf{weakly complete} if it is complete for its [[weak topology]].

\end{defn}
\begin{defn}
\label{BCplt}\hypertarget{BCplt}{}
A [[locally convex topological vector space]] is \textbf{$B$-complete} or is a \textbf{Ptak space} if a subspace of its [[continuous dual]] is [[weakly closed]] whenever its intersection with any [[equicontinuous]] subset is weakly closed (in the subset).

\end{defn}
\begin{defn}
\label{BCplt}\hypertarget{BCplt}{}
A [[locally convex topological vector space]] is \textbf{$B_r$-complete} if a dense subspace of its [[continuous dual]] is [[weakly closed]] whenever its intersection with any [[equicontinuous]] subset is weakly closed (in the subset).

\end{defn}
\begin{defn}
\label{Cplt}\hypertarget{Cplt}{}
A [[topological vector space]] is \textbf{complete} if every Cauchy [[net]] converges.

\end{defn}
\begin{defn}
\label{QuasiCplt}\hypertarget{QuasiCplt}{}
A [[topological vector space]] is \textbf{quasi-complete} if every [[bornology|bounded]], [[closed]] subset is complete.

\end{defn}
\begin{defn}
\label{SeqCplt}\hypertarget{SeqCplt}{}
A [[topological vector space]] is \textbf{sequentially complete} or \textbf{semi-complete} if every [[Cauchy sequence]] [[converges]].

\end{defn}
\begin{defn}
\label{LocCplt}\hypertarget{LocCplt}{}
A [[locally convex topological vector space]] is \textbf{locally complete} if for $B \subseteq E$ a [[bounded]], [[closed]], [[absolutely convex]] subset then its norm space, $E_B$, is a Banach space.

\end{defn}
\hypertarget{properties}{}\section*{{Properties}}\label{properties}

\hypertarget{sequentially_complete_versus_locally_complete}{}\subsection*{{Sequentially Complete versus Locally Complete}}\label{sequentially_complete_versus_locally_complete}

Sequentially complete implies locally complete because every locally Cauchy sequence is a Cauchy sequence. The inverse implication ``locally complete'' $\Rightarrow$ sequentially complete is true for example in [[metrizable]] locally convex topological vector spaces, but not in general: A Cauchy sequence will not be locally Cauchy in general.

The problem to precisly characterize the spaces in which every convergent sequence is locally convergent is an open problem according to K\"o{}the volume 1 (which is quite old, so it could have been solved in the meantime).

\hypertarget{references}{}\section*{{References}}\label{references}

See [[functional analysis bibliography]]

[[!redirects quasi-complete topological vector space]] [[!redirects semi-complete topological vector space]] [[!redirects sequentially complete topological vector space]] [[!redirects complete topological vector spaces]] [[!redirects quasi-complete topological vector spaces]] [[!redirects semi-complete topological vector spaces]] [[!redirects sequentially complete topological vector spaces]] [[!redirects complete locally convex topological vector spaces]] [[!redirects quasi-complete locally convex topological vector space]] [[!redirects semi-complete locally convex topological vector space]] [[!redirects sequentially complete locally convex topological vector space]] [[!redirects complete locally convex topological vector spaces]] [[!redirects quasi-complete locally convex topological vector spaces]] [[!redirects semi-complete locally convex topological vector spaces]] [[!redirects sequentially complete locally convex topological vector spaces]] [[!redirects complete TVS]] [[!redirects complete LCTVS]] [[!redirects quasi-complete TVS]] [[!redirects quasi-complete LCTVS]] [[!redirects semi-complete TVS]] [[!redirects semi-complete LCTVS]] [[!redirects sequentially complete TVS]] [[!redirects sequentially complete LCTVS]]



\end{document}