# Characterization of subspaces and quotients of nuclear $L_f (\alpha ,\infty)$-spaces

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Characterization of subspaces and quotients of nuclear Lf (,)-spaces COMPOSITIO MATHEMATICA HEIKKIAPIOLA Characterization of subspaces and quotients of nuclear L f (α,∞)-spaces Compositio Mathematica, tome 50, no 1 (1983), p. 65-81. <http://www.numdam.org/item?id=CM_1983__50_1_65_0> © Foundation Compositio Mathematica, 1983, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 65 CHARACTERIZATION OF SUBSPACES AND QUOTIENTS OF NUCLEAR Lf (03B1, ~)-SPACES Heikki Apiola Compositio Mathematica 50 (1983) 65-81 @ 1983 Martinus Nijhoff Publishers, The Hague. Printed in the Netherlands Introduction In a series of important papers ([15-20]) D. Vogt and M.J. Wagner study the structure of subspaces and quotients of some classes of nuclear Kôthe spaces. They are able to give a complete basis free characterization in the case of stable power series spaces thus completing and generalizing to the basis free setting the study carried out by Alpseymen, Dubinsky, Robin- son and Wagner in [2], [4], [5]-[9], [21] and [22]. The method of Vogt and Wagner can be briefly described as a combination of the following steps: (1) Proving that under suitable topological conditions concerning the Fréchet spaces E and F, a short exact sequence of the form 0 - E - G - F - 0 splits. (2) Constructing exact sequences of power series spaces of the form 0 ~ ^r(03B1) ~ ^r(03B1) ~ 0. (3) Using a suitable generalization of the Komura embed- ding theorem. The purpose of the present paper is to study to which extent this method can

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