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 $(E^{\prime}_{\bullet},d^{\prime})$ (http://planetmath.org/HomologicalComplexOfTopologicalVectorSpaces), 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.

 Title cohomological complex of topological vector spaces Canonical name CohomologicalComplexOfTopologicalVectorSpaces Date of creation 2013-03-22 18:17:27 Last modified on 2013-03-22 18:17:27 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 22 Author bci1 (20947) Entry type Definition Classification msc 55N99 Classification msc 81T70 Classification msc 32S20 Classification msc 12G10 Classification msc 55N33 Classification msc 13D25 Classification msc 18G35 Synonym cohomological complex Related topic HomologicalComplexOfTopologicalVectorSpaces Related topic ChainComplex Related topic CategoricalSequence Related topic TangentialCauchyRiemannComplexOfCinftySmoothForms Related topic ACRcomplex Defines dual of chain complex Defines cochain complex Defines transpose map Defines sequence of topological vector spaces Defines sequence of continuous linear maps