cohomological complex of topological vector spacesCohomologicalComplexOfTopologicalVectorSpaces2008-08-01 13:08:212009-06-01 00:24:06Definitiondual of chain complexcochain complextranspose mapsequence of topological vector spacessequence of continuous linear mapscohomological complexcochain complexchain complextopological vector spacescohomological complex of topological vector spaces% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\hr}{{\hookrightarrow}}\begin{definition}
A \emph{cohomological complex of topological vector spaces} is a pair $(E^{\bullet}, d)$ where
$(E^{\bullet} = (E^q)_{q \in Z} $ is a sequence of topological vector spaces and $d = (d^q)_{q \in Z }$ is
a sequence of continuous linear maps $d^q$ from $E^{q}$ into $E^{q+1}$ which satisfy
$d^q \circ d^{q+1} = 0$.
\end{definition}
\textbf{Remarks}
\begin{itemize}
\item The \emph{dual complex} of a cohomological complex $(E^{\bullet}, d)$ of topological vector spaces is the \PMlinkname{homological complex $(E'_{\bullet}, d')$}{HomologicalComplexOfTopologicalVectorSpaces}, where $(E'_{\bullet} = (E'_q)_{q \in Z}$ with $E'_q$ being the strong dual of $E^q$ and $d' = (d'_q)_{q \in Z}$ , and also with $d'_q $ being the \emph{transpose map} of $d^q$.
\item A cohomological complex of topological vector spaces (TVS) is a
specific case of a \emph{cochain complex}, which is the dual of the concept of chain complex.
\end{itemize}