DOI QR코드

DOI QR Code

BARRELLEDNESS OF SOME SPACES OF VECTOR MEASURES AND BOUNDED LINEAR OPERATORS

  • Received : 2014.10.22
  • Published : 2015.09.30

Abstract

In this paper we investigate the barrellednes of some spaces of X-valued measures, X being a barrelled normed space, and provide examples of non barrelled spaces of bounded linear operators from a Banach space X into a barrelled normed space Y, equipped with the uniform convergence topology.

Keywords

References

  1. J. Batt, P. Dierolf, and J. Voigt, Summable sequences and topological properties of $m_0$ (I), Arch. Math. (Basel) 28 (1977), no. 1, 86-90. https://doi.org/10.1007/BF01223894
  2. J. C. Diaz, M. Florencio, and P. J. Paul, A uniform boundedness theorem for $L_{\infty}({\mu},X)$, Arch. Math. (Basel) 60 (1993), no. 1, 73-78. https://doi.org/10.1007/BF01194241
  3. L. Drewnowski, M. Florencio, and P. J. Paul, The space of Pettis integrable functions is barrelled, Proc. Amer. Math. Soc. 114 (1992), no. 3, 687-694. https://doi.org/10.1090/S0002-9939-1992-1107271-2
  4. L. Drewnowski, M. Florencio, and P. J. Paul, On the barrelledness of space of bounded vector functions, Arch. Math. 63 (1994), no. 5, 449-458. https://doi.org/10.1007/BF01196676
  5. J. C. Ferrando, On the barrelledness of the vector-valued bounded function space, J. Math. Anal. Appl. 184 (1994), no. 3, 437-440. https://doi.org/10.1006/jmaa.1994.1212
  6. J. C. Ferrando, J. Kakol, and M. Lopez Pellicer, On a problem of Horvath concerning barrelled spaces of vector valued continuous functions vanishing at infinity, Bull. Belg. Math. Soc. Simon Stevin 11 (2004), no. 1, 127-132.
  7. J. C. Ferrando, M. Lopez Pellicer, and L. M. Sanchez Ruiz, Metrizable Barrelled Spaces, Pitman RNMS 332, Longman, 1995.
  8. F. J. Freniche, Barrelledness of the space of vector valued and simple functions, Math. Ann. 267 (1984), no. 4, 479-489. https://doi.org/10.1007/BF01455966
  9. H. Jarchow, Locally Convex Spaces, B. G. Teubner Stuttgart, 1981.
  10. J. Kakol, S. A. Saxon, and A. R. Todd, Barrelled spaces with(out) separable quotients, Bull. Austral. Math. Soc. 90 (2014), no. 2, 295-303. https://doi.org/10.1017/S0004972714000422
  11. G. Kothe, Topological Vector Spaces I, Springer-Verlag, New York Heidelberg Berlin, 1983.
  12. J. Mendoza, Barrelledness conditions on S (${\Sigma}$, E) and B (${\Sigma}$, E), Math. Ann. 261 (1982), no. 1, 11-22. https://doi.org/10.1007/BF01456405
  13. J. Mendoza, Necessary and sufficient conditions for C (X,E) to be barrelled or infrabarrelled, Simon Stevin 57 (1983), no. 1-2, 103-123.
  14. R. A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer, SMM, London, 2002.
  15. S. A. Saxon, Mackey hyperplanes enlargements for Tweddle's space, Rev. R. Acad. Cienc. Exactas Fis. Ser. A Math. RACSAM 108 (2014), no. 1, 1035-1054. https://doi.org/10.1007/s13398-013-0159-x
  16. S. A. Saxon, Weak barrelledness versus P-spaces, Descriptive topology and functional analysis, 2732, Springer Proc. Math. Stat., 80, Springer, Cham, 2014.
  17. J. Schmets, An example of the barrelled space associated to C (X,E), Lecture Notes in Math. 843, Functional Analysis, pp. 562-571, Holomorphy and Approximation Theory, Rio de Janeiro 1978, Springer-Verlag, 1981.