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 $({E}_{\bullet}^{\prime},{d}^{\prime})$ (http://planetmath.org/HomologicalComplexOfTopologicalVectorSpaces), where $({E}_{\bullet}^{\prime}={({E}_{q}^{\prime})}_{q\in Z}$ with ${E}_{q}^{\prime}$ being the strong dual of ${E}^{q}$ and ${d}^{\prime}={({d}_{q}^{\prime})}_{q\in Z}$ , and also with ${d}_{q}^{\prime}$ 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  20130322 18:17:27 
Last modified on  20130322 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 