Tensor sequences and inductive limits with local partition of unity
In his previous article [J. Reine Angew. Math. 319, 38-62 (1980; Zbl 0426.46053)], the author proved that a locally convex (l.c.) space E with the countable neighborhood property (c.n.p.) is an \(\epsilon\)-space if and only if \[ (+)\quad E\otimes_{\epsilon}(\sum^{\infty}_{i=1}A_ i(F_ i))\cong \sum^{\infty}_{i=1}Id\otimes A_ i(E\otimes_{\epsilon}F_ i) \] holds for all countable l.c. hulls \(F=\sum^{\infty}_{i=1}A_ i(F_ i)\) and thus characterized the spaces E for which the \(\epsilon\)-tensor product commutes with each countable l.c. hull F. Here, the author complements his discussion by characterizing those l.c. hulls F for which \((+)\) holds for all l.c. spaces E with c.n.p. In fact, this is true if and only if \(F=\sum^{\infty}_{i=1}A_ i(F_ i)\) ''admits a local partition of unity'' (a property much weaker than the existence of a partition of unity in the sense of M. de Wilde); e.g., \((+)\) holds if N(A) is an \(\epsilon\)- space or F is a \(\pi\)-space or if all \(F_ i\) are hilbertisable. As a consequence, one obtains conditions under which the \(\epsilon\)-tensor product of two bornologicalDF)-spaces is again a bornological (DF)-space; e.g., \(E\otimes_{\epsilon}F\) is a bornological (DF)-space if E is normed and F a bornological \(\pi\)-(DF)-space or if both E and F are bornological \(\pi\)-(DF)-spaces. Grothendieck's problem whether the \(\epsilon\)-tensor product of two (DF)-spaces must always be a (DF)-space remains open, but the author makes some interesting remarks on this problem and mentions several other open questions in this area. Since a l.c. hull \(F=\sum^{\infty}_{i=1}A_{\alpha}(F_{\alpha})\) has a local partition of unity if and only if \(0\to N(A)\to \oplus_{\alpha}F_{\alpha}\to F\to 0\) is a ''\(\otimes\)-sequence'', the author's work is closely related to \textit{W. Kaballo, D. Vogt} [Manuscripta Math. 32, 1-27 (1980; Zbl 0456.46058)], and he establishes two new conditions which imply that a topologically exact sequence \(0\to H\to G\to Q\to 0\) is a \(\otimes\)-sequence (viz., H is an \(\epsilon\)-space or Q is a \(\pi\)-space). The article also contains characterizations of \(\epsilon\)- and \(\pi\)-spaces in terms of \(\otimes\)-sequences and closes with applications to sequence spaces.
scientific article; zbMATH DE number 3921293
scientific article