# homological complex of topological vector spaces

###### Definition 0.1.

A homological 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 $E_{q+1}$ into $E_{q}$ which satisfy $d_{q}\circ d_{q+1}=0$.

Remarks

• The homological complex of topological vector spaces is a specifc example of a chain complex.

• A sequence of $R$-modules and their homomorphisms is said to be a $R$-complex.

 Title homological complex of topological vector spaces Canonical name HomologicalComplexOfTopologicalVectorSpaces Date of creation 2013-03-22 18:17:24 Last modified on 2013-03-22 18:17:24 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 15 Author bci1 (20947) Entry type Definition Classification msc 55N25 Classification msc 55N99 Classification msc 55R65 Classification msc 55P99 Synonym ChainComplex Related topic ChainComplex Related topic CategoricalSequence Related topic ExactSequence Related topic TangentialCauchyRiemannComplexOfCinftySmoothForms Related topic HomolgyOfMathbbRP3 Related topic CohomologicalComplexOfTopologicalVectorSpaces Related topic ACRcomplex Defines homological complex of topological vector spaces