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We say that a function f from [0,1] to a Banach space X is increasing with respect to E⊂X* if x∗∘f is increasing for every x*∈E. A function f:[0,1]m→X is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c0 or such that X* is separable, then for every separately increasing function f:[0,1]m→X with respect to any norming subset there exists a separately increasing function g:[0,1]m→R such that the sets of points of discontinuity of f and g coincide.
Słowa kluczowe
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Rocznik
Tom
Strony
61--76
Opis fizyczny
Bibliogr. 14 poz.
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autor
- Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Poznań, Poland
Bibliografia
- [1] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164.
- [2] Y. Chabrillac and J.-P. Crouzeix, Continuity and differentiability properties of monotone real functions of several real variables, in: Nonlinear Analysis and Optimization, Math. Programming Stud. 30 (1987), 1–16.
- [3] J. Ciemnoczołowski and W. Orlicz, On some classes of vector valued functions of bounded weak variation, Bull. Polish Acad. Sci. Math. 31 (1983), 335–344.
- [4] L. Drewnowski, Continuity of monotone functions with values in Banach lattices, in: Recent Progress in Functional Analysis (Valencia, 2000), North-Holland, 2001, 185–199.
- [5] L. Drewnowski and A. Michalak, Generalized Helly spaces, continuity of monotone functions and metrizing maps, Fund. Math. 200 (2008), 161–184.
- [6] G. Edgar, Measure, Topology, and Fractal Geometry, Springer, New York, 2008.
- [7] J. L. Kelley, General Topology, Van Nostrand, Toronto, 1955.
- [8] B. Lavric, A characterisation of Banach lattices with order continuous norm, Rad. Mat. 8 (1992), 37–41.
- [9] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, Berlin, 1977.
- [10] S. Łojasiewicz, An Introduction to the Theory of Real Functions, Wiley, Chichester, 1988.
- [11] A. Michalak, On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity, Studia Math. 155 (2003), 171–182.
- [12] A. Michalak, On continuous linear operators on D(0; 1) with nonseparable ranges, Comment. Math. 43 (2003), 221–248.
- [13] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1974.
- [14] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Univ. Press, Cambridge, 1991
Typ dokumentu
Bibliografia
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bwmeta1.element.baztech-2182710b-1de3-43c7-ad6d-71f990899725