Notes on Several Complex Variables: Fréchet Sheaves, Cartan a and B
نویسنده
چکیده
Throughout H(U,S) will be used for Γ(U,S) to emphasize the cohomological approach we are taking. First it might be a good idea to recall what a Fréchet space is. A seminorm is an object satisfying all of the norm axioms except for the requirement to have a nonzero kernel. A Fréchet space is a vector space F together with a sequence of seminorms {pn} on F such that (i) if pn(f) = 0 for all n, then f = 0, and (ii) if fk is a sequence that is Cauchy in every seminorm, then there exists an f such that fk → f in every seminorm. There is an induced topology given by letting the setsN(n, ) = {f ∈ F : pn(f) < } for n ∈ N and > 0 be a neighborhood basis of zero. Note that a Fréchet space is a generalization of a Banach space, which can be recovered by letting the first seminorm be an actual norm and letting all other seminorms be the zero map. Since we will need it, I will quote but not prove the following:
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