| Version 7 |
Version 6 |
| \begin{definition} |
\begin{definition} |
| A \emph{cohomological complex of topological vector spaces} is a pair $(E^{\bullet}, d)$ where |
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 |
$(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 |
a sequence of continuous linear maps $d^q$ from $E^{q}$ into $E^{q+1}$ which satisfy |
| $d^q \circ d^{q+1} = 0$. |
$d^q \circ d^{q+1} = 0$. |
| \end{definition} |
\end{definition} |
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| \textbf{Notes:} |
\textbf{Notes:} |
| \textbf{1}. The \emph{dual complex} of a cohomological complex $(E^{\bullet}, d)$ of topological vector spaces is the homological complex $(E'_{\bullet}, d')$, 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 $d^q$. \\ |
\textbf{1}. The \emph{dual complex} of a cohomological complex $(E^{\bullet}, d)$ of topological vector spaces is the homological complex $(E'_{\bullet}, d')$, 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 $d^q$. \\ |
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| \textbf{2}. A cohomological complex of topological vector spaces (TVS) is a |
\textbf{2}. A cohomological complex of topological vector spaces (TVS) is a |
| specific case of a cochain complex. |
specific case of a cochain complex. |
