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Extension of vector-valued functions. (English) Zbl 1141.46018
This is a continuation of an article by J. Bonet, L. Frerick and E. Jordá [Stud. Math. 183, No. 3, 225–248 (2007; Zbl 1141.46017)]. One of the main theorems reads as follows: Let Ω be an open and connected subset of N and a sheaf of smooth functions on Ω which is closed in the sheaf 𝒞 of continuous functions on Ω; i.e., (ω) is closed in 𝒞(ω) for each open ωΩ. Let M be a set of uniqueness for (Ω) (which means that a function in (Ω) which vanishes on M must vanish on Ω) and let E be a locally convex space. Let f:ME be a function such that uf has an extension f u (Ω) for each uWE ' . If (i) W determines boundedness, or if (ii) E is Fréchet, W= n B n , where (B n ) n fixes the topology of E (i.e., the polars (B n ) n in E form a fundamental system of 0-neighbourhoods in E), and (f u ) uB n is bounded in (Ω) for each n, then f has an extension F(Ω,E). The case (i) has already been proved in the paper by Bonet and the present authors which is quoted above.
MSC:
46E40Spaces of vector- and operator-valued functions
46A04Locally convex Fréchet spaces, etc.
46M05Tensor products of topological linear spaces