##
**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.

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
PDF
BibTeX
XML
Cite

\textit{K. D. Bierstedt} and \textit{J. Bonet}, Math. Nachr. 135, 149--180 (1988; Zbl 0688.46001)

Full Text:
DOI

### References:

[1] | Bierstedt, Transact. Amer. Math. Soc. 272 pp 107– (1982) |

[2] | , , Köthe sets and Köthe sequence spaces, pp. 27–91 in: Functional Analysis, Holomorphy and Approximation Theory, North-Holland Math. Studies 71 (1982), North-Holland, Amsterdam, New York and Oxford |

[3] | , Distinguished echelon spaces and the projective description of weighted inductive limits of type {\(\gamma\)}dC(X), pp. 169–226 in: Aspects of Mathematics and its Applications, Elsevier Science Publ., Amsterdam 1986 |

[4] | Bierstedt, Doga Tr. J. Math. 10 pp 1– (1986) |

[5] | Bonet, Collect. Math. 34 pp 117– (1983) |

[6] | Bonet, Math. Z. 192 pp 9– (1986) |

[7] | Bonet, Proc. R. Ir. Acad. 85 A pp 193– (1985) |

[8] | Defant, Manuscripta Math. 55 pp 433– (1986) |

[9] | On spaces of continuous linear mappings between locally convex spaces, Habilitationsschrift, Univ. München (1984) 73 p. |

[10] | Floret, Arch. der Math. 25 pp 646– (1974) |

[11] | Grothendieck, Summa Brasil. Math. 3 pp 57– (1954) |

[12] | Grothendieck, Mem. Amer. Math. Soc. 16 (1955) |

[13] | Heinrich, Math. Nachr. 118 pp 285– (1984) |

[14] | Math. Nachr. 121 pp 211– (1985) |

[15] | Henson, Transact. Amer. Math. Soc. 172 pp 405– (1972) |

[16] | Henson, Duke Math. J. 40 pp 193– (1973) |

[17] | Hollstein, J. reine angew. Math. 319 pp 38– (1980) |

[18] | Köthe, Grundlehren der Math. Wiss. 159 (1969) |

[19] | II., Grundlehren der Math. Wiss., vol. 237 (1979), Springer-Verlag, Berlin and New York |

[20] | Marquina, Arch. der Math. 31 pp 589– (1978) |

[21] | Mendoza Casas, Arch. der Math. 40 pp 156– (1983) |

[22] | Topis in locally convex spaces, North-Holland Math. Studies. vol. 67 (1982) North-Holland, Amsterdam, New York and Oxford |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.