, Volume 37, Issue 3, pp 269-301

Eine Charakterisierung der Potenzreihenräume von endlichem Typ und ihre Folgerungen

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Abstract

In the present article there is given a characterization of power series spaces of finite type by linear topological invariants. This is used to give a complete characterization of the classes of subspaces and quotient spaces of any nuclear stable power series space Λ1(α). It is further used for the investigation of exact sequences of the form O→Λr(α)→E→Λρ(β)→O. They are known to split for ρ=+∞. It is shown that \(E\tilde = \Lambda _r (\alpha ) \oplus \Lambda _\rho (\beta )\) for r<+∞, p<+∞, but the sequence need not split. There are examples for \(E\not \tilde = \Lambda _r (\alpha ) \oplus \Lambda _\rho (\beta )\) for r=+∞, p<+∞. At last the characterization is used to determine the structure of certain systems of linear equations in infinitely many variables (ct. Mityagin [10], Probl. 4.6). At the end of the article we give conditions on a space E for the existence of a Λ1(α) such that E is a subspace or a quotient space of E and we give an example of an isomorphy proof using the previous results.