Functiones et Approximatio Commentarii Mathematici

An operator space characterization of Fréchet spaces not containing $l^1$

Wolfgang M. Ruess

Full-text: Open access

Abstract

The classes of Fréchet spaces not containing $l^1$, of Gelfand-Phillips spaces, and of dual Gelfand-Phillips spaces are characterized by (pre)compactness criteria for sets of bounded linear operators transforming bounded sets into precompact sets.

Article information

Source
Funct. Approx. Comment. Math., Volume 44, Number 2 (2011), 259-264.

Dates
First available in Project Euclid: 22 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.facm/1308749128

Digital Object Identifier
doi:10.7169/facm/1308749128

Mathematical Reviews number (MathSciNet)
MR2841183

Zentralblatt MATH identifier
1230.46004

Subjects
Primary: 46A04: Locally convex Fréchet spaces and (DF)-spaces 46B03: Isomorphic theory (including renorming) of Banach spaces
Secondary: 46A032

Keywords
Locally convex spaces no-containment of $l^1$ limited sets precompact operators

Citation

Ruess, Wolfgang M. An operator space characterization of Fréchet spaces not containing $l^1$. Funct. Approx. Comment. Math. 44 (2011), no. 2, 259--264. doi:10.7169/facm/1308749128. https://projecteuclid.org/euclid.facm/1308749128


Export citation

References

  • N. Bourbaki, Éléments de Mathématiques, Topologie Générale, Ch. X; Hermann, Paris 1961.
  • A. Defant and K. Floret, The precompactness-lemma for sets of operators; in: Funct. Analysis, Holomorphy and Approximation Theory II (G. Zapata, Ed.), North-Holland Math. Studies 86 (1984), 39–55.
  • G. Emmanuele, A dual characterization of Banach spaces not containing $l^1$, Bull. Acad. Polon. Sci. 34 (1986), 155–160.
  • A. Grothendieck, Espaces Vectoriels Topologiques, Sociedade Mat. S. Paulo, Sao Paulo, 1964.
  • R.H. Lohman, On the selection of basic sequences in Fréchet spaces, Bull. Inst. Math. Acad. Sinica 2 (1974), 145–152.
  • F. Mayoral, Compact sets of compact operators in absence of $l^1$, Proc. Amer. Math. Soc. 129 (2000), 79–82.
  • W.M. Ruess, Compactness and collective compactness in spaces of compact operators, J. Math. Anal. Appl. 84 (1981), 400–417.
  • H.P. Rosenthal, A characterization of Banach spaces containing $l^1$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413.
  • Th. Schlumprecht, On limitedness in locally convex spaces, Arch. Math. 53 (1989), 65–74.
  • L. Schwartz, Théorie des distributions à valeurs vectorielles (I), Ann. Inst. Fourier 7 (1957), 1–141.
  • M. Valdivia, Fréchet spaces with no subspaces isomorphic to $ l_1$, Math. Japonica 38 (1993), 397–411.