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Twisted sums of Banach and nuclear spaces. (English) Zbl 0595.46002

A twisted sum of topological vector spaces Y and Z is a space X with a subspace \(Y_ 1\) isomorphic to Y for which \(X/Y_ 1\) is isomorphic to Z. It splits if \(Y_ 1\) is complemented. It is proved directly that every twisted sum of a Banach space Y and a nuclear space Z splits [another independent proof: D. Vogt, Some results on continuous linear maps between Fréchet spaces, Functional analysis: Surveys and recent results III, North-Holland Math. Stud. 90, 349-381 (1984; Zbl 0585.46060)] and that every locally convex twisted sum of a nuclear Fréchet space Y and a Banach space Z splits too [another proof: V. P. Palamodov, Usp. Mat. Nauk 26, No.1(157), 3-65 (1971; Zbl 0247.46070)]. Köthe sequence spaces Z for which very twisted sum of any Banach space Y and Z splits are characterized.

MSC:

46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
46A45 Sequence spaces (including Köthe sequence spaces)
46M10 Projective and injective objects in functional analysis
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