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Riemann step function approximation of Bochner integrable functions. (English) Zbl 0595.28013
Given a Banach space \((X,\| \cdot \|)\), consider the space \(L^ 1(0,T;X)\) of all X-valued, Bochner integrable functions f defined a.e. on the interval \([0,T],\) and having norm \(\| f\|_{L^ 1(0,T;X)}=\int^{T}_{0}\| f(t)\| dt.\) The author shows that f is the uniform limit in the \(L^ 1\)-norm of its Riemann step function approximations along nearly every sequence of partitions of \([0,T]\) with mesh size approaching zero.
Reviewer: O.Lipovan
MSC:
28B05 Vector-valued set functions, measures and integrals
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[1] H. Brézis, Opérateurs maximaux monotones, North-Holland, Amsterdam, 1973. · Zbl 0252.47055
[2] M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math. 11 (1972), 57 – 94. · Zbl 0249.34049 · doi:10.1007/BF02761448 · doi.org
[3] L. C. Evans, Nonlinear evolution equations in an arbitrary Banach space, Israel J. Math. 26 (1977), no. 1, 1 – 42. · Zbl 0349.34043 · doi:10.1007/BF03007654 · doi.org
[4] M. A. Freedman, Product integrals of continuous resolvents: existence and nonexistence, Israel J. Math. 46 (1983), no. 1-2, 145 – 160. · Zbl 0537.34060 · doi:10.1007/BF02760628 · doi.org
[5] M. A. Freedman, Existence of strong solutions to singular nonlinear evolution equations, Pacific J. Math. 120 (1985), no. 2, 331 – 344. · Zbl 0592.34042
[6] Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. · Zbl 0078.10004
[7] Tosio Kato, Linear evolution equations of ”hyperbolic” type. II, J. Math. Soc. Japan 25 (1973), 648 – 666. · Zbl 0262.34048 · doi:10.2969/jmsj/02540648 · doi.org
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