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Convex transformations with Banach lattice range. (English) Zbl 0675.46012

A proof is given of a convex analogue of the Banach closed graph theorem, to the effect that if X is a real linear topological Baire space, Y is a Banach lattice with strong unit, D a non-empty open convex subset of X and f:D\(\to Y\) satisfies
(i) f(\(\frac{x+y}{2})\leq \frac{f(x)+f(y)}{2}(x,y\in D)\)
(ii) \(\{\) (x,y)\(\in D\times Y:f(x)\leq y\}\) is closed in \(D\times Y,\)
then f is continuous.
A number of corollaries is presented, related in particular to the work of Ih-Ching Hsu [Proc. Am. Math. Soc. 58, 119-123 (1976; Zbl 0334.46009)].
Reviewer: C.B.Huijsmans

MSC:

46B42 Banach lattices
46A40 Ordered topological linear spaces, vector lattices
26A51 Convexity of real functions in one variable, generalizations

Citations:

Zbl 0334.46009
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