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cohomological complex of topological vector spaces∗
bci1†
2013-03-22 1:29:53
Definition 0.1. A cohomological complex of topological vector spaces is a pair (E • , d) where (E • = (E q )q∈Z
is a sequence of topological vector spaces and d = (dq )q∈Z is a sequence of continuous linear maps dq from
E q into E q+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•0 , d0 ), where (E•0 = (Eq0 )q∈Z with Eq0 being the strong dual of E q and d0 = (d0q )q∈Z , and
also with d0q 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.
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