Abstract
Let S(μ, E) be the space of (classes of μ-a.e. equal) simple functions defined on a (non-trivial) measure space with values in a locally convex space E. The following results hold: S(μ,E) is quasi-barrelled (resp. bornological) if and only if E is quasi-barrelled (resp. bornological) and E′(β(E′,E)) has the property (B) of Pietsch; S(μ, E) is barrelled if and only if S(μ,K) is barrelled and E is barrelled and nuclear; S(μ, E) is never ultrabornological; and S(μ, E) is a DF-space if and only if E is a DF-space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Batt, P. Dierolf and J. Voigt, Summable sequences and topological properties of m0(I), Arch. Math. (Basel), 28 (1977), 86–90.
A. Defant and W. Govaerts, Bornological and ultrabornological spaces of type C(X, F) and EεF, Math. Ann., 268 (1984), 347–355.
A. Defant and W. Govaerts, Tensor products and spaces of vector-valued continuous functions, Manuscripta Math., 55 (1986), 433–449.
P. Dierolf, S. Dierolf and L. Drewnowski, Remarks and examples concerning unordered Baire-like and ultrabarrelled spaces, Colloq. Math., 39 (1978), 109–116.
N. Dinculeanu, Vector Measures, Pergamon Press, Oxford, 1967.
F. J. Freniche, Barrelledness of the space of vector-valued and simple functions, Math. Ann., 267 (1984), 479–486.
R. Hollstein, Inductive limits and e-tensor products, J. Reine Angew. Math., 319 (1980), 38–62.
H. Jarchow, Locally Convex Spaces, B. G. Teubner, Stuttgart, 1981.
G. Köthe, Topological Vector Spaces II, Springer-Verlag, Berlin, Heidelberg and New York, 1979.
A. Marquina and J. M. Sanz-Serna, Barrelledness conditions on c0(E), Arch. Math. (Basel), 31 (1978), 589–596.
J. Mendoza, Barrelledness conditions on S(∑, E) and B(∑, E), Math. Ann., 261 (1982), 11–22.
P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, Notas Mat., vol. 131, North-Holland, Amsterdam, New York and Oxford, 1987.
A. Pietsch, Nuclear Locally Convex Spaces, Springer-Verlag, Berlin, Heidelberg and New York, 1972.
J. Schmets, Spaces of Vector-valued Continuous Functions, Lecture Notes in Math., vol. 1003, Springer-Verlag, Berlin, Heidelberg and New York, 1983.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research has been supported by La Consejería de Education y Ciencia de la Junta de Andalucía.
Rights and permissions
About this article
Cite this article
Diaz, S., Drewnowski, L., Fernandez, A. et al. Barrelledness and bornological conditions on spaces of vector-valued μ-simple functions. Results. Math. 21, 289–298 (1992). https://doi.org/10.1007/BF03323086
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03323086