Domański, Paweł Twisted sums of Banach and nuclear spaces. (English) Zbl 0595.46002 Proc. Am. Math. Soc. 97, 237-243 (1986). 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. Cited in 2 Documents 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 Keywords:complemented subspace; lifting; twisted sum of topological vector spaces; every twisted sum of a Banach space Y and a nuclear space Z splits; every locally convex twisted sum of a nuclear Fréchet space Y; and a Banach space Z splits too; Köthe sequence spaces; every locally convex twisted sum of a nuclear Fréchet space Y and a Banach space Z splits too Citations:Zbl 0585.46060; Zbl 0247.46070 PDFBibTeX XMLCite \textit{P. Domański}, Proc. Am. Math. Soc. 97, 237--243 (1986; Zbl 0595.46002) Full Text: DOI