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Tensor sequences and inductive limits with local partition of unity. (English) Zbl 0576.46053
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 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.
Reviewer: K.D.Bierstedt

46M05 Tensor products in functional analysis
46M40 Inductive and projective limits in functional analysis
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46A08 Barrelled spaces, bornological spaces
46F05 Topological linear spaces of test functions, distributions and ultradistributions
46A45 Sequence spaces (including Köthe sequence spaces)
46B25 Classical Banach spaces in the general theory
46A04 Locally convex Fréchet spaces and (DF)-spaces
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