PlanetPhysics/Cohomological Complex

A cohomological complex of \htmladdnormallink{topological {http://planetphysics.us/encyclopedia/CoIntersections.html} 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 \htmladdnormallink{homological complex $$(E'_{\bullet}, d')$$}{http://planetphysics.us/encyclopedia/HomologicalComplexOfTopologicalVectorSpaces.html}, 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 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.