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Stefan Heinrich’s density condition for Fréchet spaces and the characterization of the distinguished Köthe echelon spaces. (English) Zbl 0688.46001
In connection with his study of ultrapowers of locally convex spaces S. Heinrich introduced the “density condition”: Let E denote a locally convex space, \({\mathfrak U}(E)\) the system of all closed absolutely convex neighborhoods of 0 in E, and \({\mathfrak B}(E)\) the system of all closed absolutely convex and bounded subsets of E. Then E is defined to satisfy the density condition (DC), if the following holds: Given any function \(\lambda\) : \({\mathfrak U}(E)\to {\mathbb{R}}_+\setminus \{0\}\) and an arbitrary \(V\in {\mathfrak U}(E)\), there always exist a finite set \({\mathfrak U}\subset {\mathfrak U}(E)\) and \(B\in {\mathfrak B}(E)\) such that \(\cap \{\lambda (U)U:\) \(U\in {\mathfrak U}\}\subset B+V\) [Math. Nachr. 118, 285-315 (1989; Zbl 0576.46002), Math. Nachr. 121, 211-229 (1985; Zbl 0601.46001)].
In the first section the authors show that for a metrizable locally convex space E the following assertions are equivalent:
(1) E satisfies DC,
(2) each bounded set of the strong dual \(E'_ b\) is metrizable,
(3) \(\ell^ 1(E)\) is distinguished,
(4) \(\ell^{\infty}(E'_ b)\) \((\cong \ell_ 1(E)'_ b)\) is barrelled (bornological),
(5) \(\ell^ 1(E)\) satisfies DC.
From the equivalence of (1) and (2) and a well known theorem of Grothendieck on (DF)-spaces, it is clear that an (F)-space satisfyng DC is distinguished. At the end of the paper an example of a distinguished (F)-space (even with separable dual) is given, which does not satisfy DC.
In the second section some interesting new results on Köthe echelon spaces are proved. Let \(\lambda_ 1=\lambda_ 1(I,A)\) be an echelon space of oder 1 with index set I and Köthe matrix A. In an earlier paper [Aspects of mathematics and its applications, 169-226 (1986; Zbl 0645.46027)] the first named author and R. Meise introduced some condition D on A, which is sufficient for \(\lambda_ 1\) to be distinguished. It is proved, that \(\lambda_ 1\) satisfies DC iff A satisfies D, from which now the equivalence of D with the distinguishedness of \(\lambda_ 1\) can be deduced. As an application the distinguished vector valued echelon spaces \(\lambda_ 1(E)\), E (F)- space, are characterized. For echelon spaces \(\lambda_ p=\lambda_ p(I,A)\) of order p, \(1\leq p<\infty\) or \(p=0\), the main result is the equivalence of the following assertions:
(1) A satisfies D,
(2) \(\lambda_ p\) satisfies DC,
(3) \(\ell_ 1{\hat \oplus}_{\pi}\lambda_ p\) \((\cong \ell^ 1(\lambda_ p))\) is distinguished,
(4) \(\lambda_ 1{\hat \oplus}_{\pi}\lambda_ p\) \((\cong \lambda (\lambda_ p))\) is distinguished for every echelon space \(\lambda_ 1\) of order 1.
As a consequence of their results the authors finally obtain important new insight in Köthe-Grothendieck’s and Amemiya’s examples. There are also some interesting results on vector valued echelon spaces.
Reviewer: H.Pfister

MSC:
46A04 Locally convex Fréchet spaces and (DF)-spaces
46A45 Sequence spaces (including Köthe sequence spaces)
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