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.