PlanetPhysics/Cohomological Complex

A cohomological complex of \htmladdnormallink{topological {http://planetphysics.us/encyclopedia/CoIntersections.html} [[../NormInducedByInnerProduct/|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 $(E{\text{'}}_{\bullet },d^{\prime })$}{http://planetphysics.us/encyclopedia/HomologicalComplexOfTopologicalVectorSpaces.html}, where $(E{\text{'}}_{\bullet }=(E{\text{'}}_q{)}_{q\in Z}$ with $E{\text{'}}_q$ being the strong dual of $E^q$ and $d^{\prime }=(d{\text{'}}_q{)}_{q\in Z}$ , and also with $d{\text{'}}_q$ 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 [[../PreciseIdea/|concept]] of chain complex.