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 $d={(d^q)}_{q\in Z}$ isa 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 aspecific 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