Skip to main content
SpringerLink
Log in

Projective Descriptions of Weighted Inductive Limits: The Vector-Valued Cases

  • Chapter
Advances in the Theory of Fréchet Spaces

Part of the book series: NATO ASI Series ((ASIC,volume 287))

Abstract

This is a report on some recent research on the projective description for weighted inductive limits of spaces of vector-valued continuous functions and on topological properties of vector-valued echelon and co-echelon spaces. In the first part, it is pointed out that most of the theorems known in the scalar case remain valid if the functions take values in a (DF)-space and that, similarly, many properties of the classical sequence spaces carry over to echelon spaces with values in a Fréchet space resp. to (DF)-space-valued co-echelon spaces. In the second part, we turn to the case that the continuous functions take their values in Fréchet spaces. This is related to some properties in the structure theory of Fréchet spaces and work by D. Vogt; part of the results in this direction are due to A. Galbis.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. F. Bastin, ‘On bornological spaces CV(X)’, to appear in Arch. Math.

    Google Scholar 

  2. E. Behrends, S. Dierolf, P. Harmand, ‘On a problem of Bellenot and Dubinsky’, Math. Ann. 275 (1986) 337–339.

    Article  MathSciNet  MATH  Google Scholar 

  3. S.F. Bellenot, E. Dubinsky, ‘Fréchet spaces with nuclear Köthe quotients’, Trans. Amer. Math. Soc. 273 (1982) 579–594.

    MathSciNet  MATH  Google Scholar 

  4. C. Bessaga, A. Pełczyński, S. Rolewicz, ‘On diametral approximative dimension and linear homogeneity of F-spaces’, Bull. Acad. Polon. Sci. 9 (1961) 677–683.

    MATH  Google Scholar 

  5. K. D. Bierstedt, J. Bonet, ‘Stefan Heinrich’s density condition for Frechet spaces and the characterization of the distinguished Köthe echelon spaces’, Math. Nachr. 135 (1988) 149–180.

    Article  MathSciNet  MATH  Google Scholar 

  6. K.D. Bierstedt, J. Bonet, ‘Dual density conditions in (DF)-spaces I’, Results Math. 14 (1988) 242–274.

    MathSciNet  MATH  Google Scholar 

  7. K.D. Bierstedt, J. Bonet, ‘Dual density conditions in (DF)-spaces II’, to appear in Bull. Soc. Roy. Sci. Liège (1988).

    Google Scholar 

  8. KJD. Bierstedt, J. Bonet, J. Schmets, ‘(DF)-spaces of type CB(X, E) and CV(X, E)’, preprint 1988.

    Google Scholar 

  9. K.D. Bierstedt, R. Meise,‘Induktive Limites gewichteter Räume stetiger und holomorpher Funktionen’, J. reine angew. Math. 282 (1976) 186–220.

    MathSciNet  MATH  Google Scholar 

  10. K.D. Bierstedt, R. Meise, ‘Distinguished echelon spaces and projective descriptions of weighted inductive limits of type VC(X)’, pp.169–226 in: Aspects of Mathematics and its Applications, Elsevier Sci. Publ., North-Holland Math. Library, 1986.

    Google Scholar 

  11. K.D. Bierstedt, R. Meise, ‘Weighted inductive limits and their projective descriptions’, Doga Tr. J. Math. 10, 1 (1986) (Special issue: Proceedings of the Silviri Conference 1985) 54–82.

    MathSciNet  MATH  Google Scholar 

  12. K.D. Bierstedt, R. Meise, W. H. Summers, ‘A projective description of weighted inductive limits’, Trans. Amer. Math. Soc. 272 (1980) 107–160.

    Article  MathSciNet  Google Scholar 

  13. K.D. Bierstedt, R. Meise, W. H. Summers, ‘Köthe sets and Köthe sequence spaces’, pp. 27–91 in: Functional Analyis, Holomorphy and Approximation Theory, North-Holland Math. Studies 71, 1982.

    Google Scholar 

  14. J. Bonet, ‘A projective description of weighted inductive limits of spaces of vector valued continuous functions’, Collectanea Math. 34 (1983) 115–125.

    MathSciNet  Google Scholar 

  15. J. Bonet, ‘The countable neighbourhood property and tensor products’, Proc. Edinburgh Math. Soc. 28 (1985) 207–215.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Bonet, ‘On weighted inductive limits of spaces of continuous functions’, Math. Z. 192 (1986) 9–20.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Bonet, ‘Projective descriptions of inductive limits of Freenet sequence spaces’, Arch. Math. 48 (1987) 331–336.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Bonet, A. Galbis, ‘A note on Taskinen’s counterexample to the problem of topologies of Grothendieck’, to appear in Proc. Edinburgh Math. Soc.

    Google Scholar 

  19. J. Bonet, A. Galbis, ‘The identity L(E, F)=LB(E, F), tensor products and inductive limits’, to appear in Note di Mat.

    Google Scholar 

  20. J. Bonet, M. Maestre, G. Metafune, V. B. Moscatelli, D. Vogt, ‘Every quojection is the quotient of a product of Banach spaces’, preprint 1988.

    Google Scholar 

  21. A. Defant, K. Floret, ‘The precompactness-lemma for sets of operators’, pp. 39–55 in: Functional Analysis, Holomorphy and Approximation Theory II, North-Holland Math. Studies 86, 1984.

    Chapter  Google Scholar 

  22. A. Defant, W. Govaerts, ‘Tensor products and spaces of vector-valued continuous functions’, Manuscr. Math. 55 (1986) 433–449.

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Dierolf, ‘On spaces of continuous linear mappings between locally convex spaces’, Note di Mat. 5 (1985) 147–255.

    MathSciNet  MATH  Google Scholar 

  24. K. Floret, ‘Some aspects of the theory of locally convex inductive limits’, pp. 205–237 in: Functional Analysis, Surveys and Recent Results II, North-Holland Math. Studies 38, 1980.

    Chapter  Google Scholar 

  25. A. Galbis, ‘Commutatividad entre límites inductivos y productos tensoriales’, Thesis, University of Valencia, 1988.

    Google Scholar 

  26. A. Galbis, ‘Köthe sequence spaces with values in Fréchet or (DF)-spaces’, Bull. Soc. Roy. Sci. Liège 57 (1988) 157–172.

    MathSciNet  MATH  Google Scholar 

  27. A. Galbis, ‘Projective descriptions of weighted inductive limits of spaces of continuous functions with values in a Freenet space’, preprint 1988.

    Google Scholar 

  28. A. Grothendieck, ‘Sur les espaces (F) et (DF)’, Summa Brasil. Math. 3 (1954) 57–112.

    MathSciNet  Google Scholar 

  29. A. Grothendieck, ‘Produits tensoriels topologiques et espaces nucléaires’, Mem. Amer. Math. Soc. 16, 1955.

    Google Scholar 

  30. R. Hollstein, ‘Inductive limits and e-tensor products’, J. reine angew. Math. 319 (1980) 38–62.

    Article  MathSciNet  MATH  Google Scholar 

  31. H. Jarchow, ‘Locally Convex Spaces’, B.G. Teubner, Stuttgart, 1981.

    MATH  Google Scholar 

  32. J. Krone, D. Vogt, ‘The splitting relation for Köthe spaces’, Math. Z. 190 (1985) 387–400.

    Article  MathSciNet  MATH  Google Scholar 

  33. V. B. Moscatelli: ‘Fréchet spaces without continuous norms and without bases’, Bull. London Math. Soc. 12 (1980) 63–66.

    Article  MathSciNet  MATH  Google Scholar 

  34. P. Pérez Carreras, J. Bonet, ‘Barrelled Locally Convex Spaces’, North-Holland Math. Studies 131, 1987.

    MATH  Google Scholar 

  35. J. Taskinen, ‘Counterexamples to “problème des topologies” of Grothendieck’, Ann. Acad. Sci. Fenn., Serie A, no 63, 1986.

    MATH  Google Scholar 

  36. J. Taskinen, The projective tensor product of Fréchet-Montel spaces’, Studia Math. 91 (1988) 17–30.

    MathSciNet  MATH  Google Scholar 

  37. D. Vogt: ‘On the functor Ext1(E, F) for Fréchet spaces’, Studia Math. 85 (1987) 163–197.

    MathSciNet  Google Scholar 

  38. D. Vogt, Frécheträume, zwischen denen jede stetige lineare Abbildung beschränkt ist’, J. reine angew. Math. 345 (1983) 182–200.

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Vogt, ‘Some results on continuous linear maps between Fréchet spaces’. pp. 349–381 in Functional Analysis: Surveys and Recent Results III, North-Holland Math. Studies 90, 1984.

    Chapter  Google Scholar 

  40. D. Vogt, ‘On two problems of Mityagin’, preprint, Wuppertal 1987.

    Google Scholar 

  41. D. Vogt, ‘Distinguished Köthe spaces’, to appear in Math. Z.

    Google Scholar 

  42. D. Vogt, ‘On two classes of (F)-spaces’, Arch. Math. 45 (1985) 225–266.

    Google Scholar 

  43. D. Vogt, ‘Lectures on projective spectra of (DF)-spaces’, Wuppertal 1987.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. FB 17, Univ.-GH-Paderborn, Postfach 1621, D-4790, Paderborn, F.R. of Germany

    Klaus D. Bierstedt

  2. Matemáticas E.T.S Arquitectura, Univ. Politécnica, E-46071, Valencia, Spain

    José Bonet

Authors
  1. Klaus D. Bierstedt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José Bonet
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Middle East Technical University, Ankara, Turkey

    T. Terzioñlu

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Kluwer Academic Publishers

About this chapter

Cite this chapter

Bierstedt, K.D., Bonet, J. (1989). Projective Descriptions of Weighted Inductive Limits: The Vector-Valued Cases. In: Terzioñlu, T. (eds) Advances in the Theory of Fréchet Spaces. NATO ASI Series, vol 287. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2456-7_12

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-2456-7_12

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7608-1

  • Online ISBN: 978-94-009-2456-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation