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

DOI:
https://doi.org/10.1090/S0002-9939-1974-0423076-9

MathSciNet review:
0423076

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Abstract: Let be a Banach lattice, and be an open interval on the real line. A function is defined to be weakly convex if there exists a nonnegative nondecreasing continuous function such that , whenever and are in for each positive linear functional on . A representation theorem is proved as follows: If is weakly convex on and is bounded on an interval contained in , then , where is the Bochner integral of on with and .

**[1]**G. H. Hardy, J. E. Littlewood and G. Pólya,*Inequalities*, 2nd ed., Cambridge Univ. Press, London and New York, 1952. MR**13**, 727. MR**0046395 (13:727e)****[2]**I. Hsu,*Weak convexity of operator-valued functions*(unpublished).**[3]**G. Birkhoff,*Lattice theory*, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R. I., 1967. MR**37**#2638. MR**0227053 (37:2638)****[4]**K. Yosida,*Functional analysis*, Die Grundlehren der math. Wissenschaften, Band 123, Springer-Verlag, Berlin and New York, 1965. MR**31**#5054.**[5]**G. Jameson,*Ordered linear spaces*, Springer-Verlag, Berlin and New York, 1970. MR**0438077 (55:10996)****[6]**I. Hsu,*A functional inequality and its relation to convexity of vector-valued functions*(submitted).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0423076-9

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