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Simultaneous smoothing and interpolation with respect to E. Borel’s theorem. (English) Zbl 0781.46002
The following tame splitting theorem for short exact sequences of Fréchet spaces is proved: Every tamely exact sequence of the form \[ 0 \longrightarrow E \longrightarrow G\longrightarrow \Lambda^ 1_ \infty(\alpha) \longrightarrow 0 \] splits tamely if \(E\) satisfies a condition called \((\Omega DS)\) which is slightly stronger than property \((\Omega)\) in standard form introduced by D. Vogt und M. J. Wagner; here \(\Lambda^ 1_ \infty(\alpha)\) denotes a power series space of infinite type. In contrast to known results, the above theorem needs neither nuclearity nor the assumption that the seminorms are induced by semiscalar products. The proof is based on a result which combines a classical theorem of E. Borel with the technique of smoothing a function by convolution; this result is used for a verification of a technical condition due to D. Vogt which is sufficient for a time version of \(\text{Proj}^ 1 {\mathcal X}=0\) in the sense of the projective limit functor of V. P. Palamodov.

MSC:
46A04 Locally convex Fréchet spaces and (DF)-spaces
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