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cohomological complex of topological vector spaces
Definition 0.1.
A 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 $d^{q}\circ d^{{q+1}}=0$.
Remarks

The dual complex of a cohomological complex $(E^{{\bullet}},d)$ of topological vector spaces is the homological complex $(E^{{\prime}}_{{\bullet}},d^{{\prime}})$, where $(E^{{\prime}}_{{\bullet}}=(E^{{\prime}}_{q})_{{q\in Z}}$ with $E^{{\prime}}_{q}$ being the strong dual of $E^{q}$ and $d^{{\prime}}=(d^{{\prime}}_{q})_{{q\in Z}}$ , and also with $d^{{\prime}}_{q}$ being the transpose map of $d^{q}$.

A cohomological complex of topological vector spaces (TVS) is a specific case of a cochain complex, which is the dual of the concept of chain complex.
Mathematics Subject Classification
55N99 no label found81T70 no label found32S20 no label found12G10 no label found55N33 no label found13D25 no label found18G35 no label found Forums
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