Definition 0.1.A cohomological complex of topologicalvector spaces is a pair (E∙,d) where
(E∙ = (Eq)q∈Z is a sequence of topological vector spaces and d = (dq)q∈Z is a sequence of
continuous linear maps dq from Eq into Eq+1 which satisfy dq∘ dq+1 = 0.
Remarks
The dual complex of a cohomological complex (E∙,d) of topological vector spaces is
the homological complex (E′∙,d′), where (E′∙ = (E′q)q∈Z with E′q being the strong
dual of Eq and d′ = (d′q)q∈Z , and also with d′q being the transpose map of dq.
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.
