Abstract
We show that although the differentiability and the weak differentiability are different “locally”, these notions coincide almost everywhere “globally”. It is proved that a Banach space valued function F : [0, 1] → X is differentiable almost everywhere on [0, 1], if and only if F is weakly differentiable almost everywhere on [0, 1].
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References
Bongiorno D.: Stepanoff’s theorem in separable Banach spaces. Comment. Math. Univ. Carolinae 39, 323–335 (1998)
Diestel, J., Uhl, J.J.: Vector Measures. Math. Surveys, vol.15, Amer. Math. Soc., Providence (1977)
Dunford N., Schwartz J.T.: Liner Operators, Part I: General Theory. Interscience, New York (1958)
Kaliaj S.B.: The average range characterization of the Radon–Nikodym property. Mediterr. J. Math. 11(3), 905–911 (2014)
Kaliaj, S.B.: Some full characterizations of differentiable functions. Mediterr. J. Math. doi:10.1007/s00009-014-0458-2. Springer, Basel (2014)
Kaliaj S.B.: The differentiability of Banach space-valued functions of bounded variation. Monatsh. Math. 173(3), 343–359 (2014)
Naralenkov K.: On Denjoy type extensions of the Pettis integral. Czechoslovak Math. J. 60(3), 737–750 (2010)
Yosida K.: Functional Analysis. Springer, Berlin (1980)
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Kaliaj, S.B. Differentiability and Weak Differentiability. Mediterr. J. Math. 13, 2801–2811 (2016). https://doi.org/10.1007/s00009-015-0656-6
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DOI: https://doi.org/10.1007/s00009-015-0656-6