Stefan Heinrich’s density condition for Fréchet spaces and the characterization of the distinguished Köthe echelon spaces. (English) Zbl 0688.46001

Bierstedt, Klaus Dieter; Bonet, José

Math. Nachr. 135, 149-180 (1988).

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)

Keywords:

Fréchet spaces and their duals; Köthe echelon spaces; density condition; metrizable locally convex; distinguished; barrelled; bornological

Citations:

Zbl 0576.46002; Zbl 0601.46001; Zbl 0645.46027

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