Definition 0.1. A cohomological complex of topological vector 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.
