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Proceedings of the American Mathematical Society

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Convexity of vector-valued functions

Authors: Ih Ching Hsu and Robert G. Kuller
Journal: Proc. Amer. Math. Soc. 46 (1974), 363-366
MSC: Primary 46G99; Secondary 26A51
MathSciNet review: 0423076
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Abstract: Let $ (\mathcal{B}, \ll )$ be a Banach lattice, and $ (a,b)$ be an open interval on the real line. A function $ F:(a,b) \to \mathcal{B}$ is defined to be weakly convex if there exists a nonnegative nondecreasing continuous function $ G:(a,b) \to \mathcal{B}$ such that $ p[F(S)] + tp[G(s)] \leq p[F(s + t)]$, whenever $ s$ and $ s + t$ are in $ (a,b)$ for each positive linear functional $ p$ on $ \mathcal{B}$. A representation theorem is proved as follows: If $ F$ is weakly convex on $ (a,b)$ and is bounded on an interval contained in $ (a,b)$, then $ (B)\int_{a + \epsilon }^x {G(s)dm = F(x) - F(a + \epsilon )} $, where $ (B)\int_{a + \epsilon }^x {G(s)dm} $ is the Bochner integral of $ G$ on $ [a, \epsilon ,x]$ with $ 0 < \epsilon $ and $ a < \epsilon < x < b$.

References [Enhancements On Off] (What's this?)

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Keywords: Weak convexity, Banach lattice, positive linear functional, Bochner integral, order-bounded, metric-bounded, Riesz decomposition theorem, weakly Lebesgue integrable, strongly Lebesgue measurable, Hahn-Banach theorem, strong convexity
Article copyright: © Copyright 1974 American Mathematical Society