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Barrelledness and bornological conditions on spaces of vector-valued \(\mu\)-simple functions. (English) Zbl 0813.46001

The space \(S(\mu, E)\) of classes of \(\mu\)-a.e.-equal simple functions defined on a non-trivial measure space with values in a locally convex space \(E\) is considered with the topology of uniform convergence. It is shown that \(S(\mu, E)\) is quasi-barrelled (resp. bornological) if and only if \(E\) is of the same type and \(E_ \beta^ \prime \) has property \((B)\) of Pietsch. Furthermore \(S(\mu, E)\) is barrelled (resp. DF) if and only if \(E\) is barrelled and nuclear (resp. \(E\) is DF). The space \(S(\mu, E)\) is never ultrabornological. The proofs are based on the isomorphism \(E(\mu, E)\cong S(\mu, \mathbb{K})\otimes_ \varepsilon E\) and on Stone’s representation theorem.
Reviewer: A.Kriegl (Wien)

MSC:

46A08 Barrelled spaces, bornological spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
28B05 Vector-valued set functions, measures and integrals
46M05 Tensor products in functional analysis
46A32 Spaces of linear operators; topological tensor products; approximation properties
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[1] J. Batt, P. Dierolf and J. Voigt, Summable sequences and topological properties of m0(I), Arch. Math. (Basel), 28 (1977), 86–90. · Zbl 0362.46007 · doi:10.1007/BF01223894
[2] A. Defant and W. Govaerts, Bornological and ultrabornological spaces of type C(X, F) and E{\(\epsilon\)}F, Math. Ann., 268 (1984), 347–355. · Zbl 0525.46024 · doi:10.1007/BF01457063
[3] A. Defant and W. Govaerts, Tensor products and spaces of vector-valued continuous functions, Manuscripta Math., 55 (1986), 433–449. · Zbl 0601.46064 · doi:10.1007/BF01186656
[4] P. Dierolf, S. Dierolf and L. Drewnowski, Remarks and examples concerning unordered Baire-like and ultrabarrelled spaces, Colloq. Math., 39 (1978), 109–116. · Zbl 0386.46008
[5] N. Dinculeanu, Vector Measures, Pergamon Press, Oxford, 1967.
[6] F. J. Freniche, Barrelledness of the space of vector-valued and simple functions, Math. Ann., 267 (1984), 479–486. · Zbl 0525.46022 · doi:10.1007/BF01455966
[7] R. Hollstein, Inductive limits and e-tensor products, J. Reine Angew. Math., 319 (1980), 38–62. · Zbl 0426.46053
[8] H. Jarchow, Locally Convex Spaces, B. G. Teubner, Stuttgart, 1981.
[9] G. Köthe, Topological Vector Spaces II, Springer-Verlag, Berlin, Heidelberg and New York, 1979. · Zbl 0417.46001
[10] A. Marquina and J. M. Sanz-Serna, Barrelledness conditions on c0(E), Arch. Math. (Basel), 31 (1978), 589–596. · Zbl 0387.46004 · doi:10.1007/BF01226496
[11] J. Mendoza, Barrelledness conditions on S( E) and B( E), Math. Ann., 261 (1982), 11–22. · Zbl 0477.46030 · doi:10.1007/BF01456405
[12] P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, Notas Mat., vol. 131, North-Holland, Amsterdam, New York and Oxford, 1987. · Zbl 0614.46001
[13] A. Pietsch, Nuclear Locally Convex Spaces, Springer-Verlag, Berlin, Heidelberg and New York, 1972. · Zbl 0308.47024
[14] J. Schmets, Spaces of Vector-valued Continuous Functions, Lecture Notes in Math., vol. 1003, Springer-Verlag, Berlin, Heidelberg and New York, 1983. · Zbl 0511.46033
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