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On the splitting relation for Fréchet-Hilbert spaces. (English) Zbl 1230.46005

The splitting problem for exact sequences of locally convex spaces consists in characterizing those pairs \((E,F)\) of locally convex spaces for which every short exact sequence \(0\rightarrow F\rightarrow G\rightarrow E \rightarrow 0\) splits. For pairs \((E,F)\) of Fréchet-Hilbert spaces, this was solved by P. Domański and M. Mastyło, [“Characterization of splitting for Fréchet–Hilbert spaces via interpolation”, Math. Ann. 339, No. 2, 317–340 (2007; Zbl 1132.46043)]. In the present paper, a short elegant proof of this result is presented, including a new formulation and the proof of the crucial interpolation theorem from [loc. cit.] and full proofs of the results on acyclicity of inductive spectra, as far as they are needed in the difficult sufficiency part of the characterization.

MSC:

46A04 Locally convex Fréchet spaces and (DF)-spaces
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
46M40 Inductive and projective limits in functional analysis
46A63 Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces
46B70 Interpolation between normed linear spaces

Citations:

Zbl 1132.46043
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References:

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