Summary
It is shown that for any real Baire topological vector spaceX the set classesA(X):={T ⊂ χ: for any open and convex setD ⊃ T, every Jensen-convex functional, defined onD and bounded from above onT, is continuous} andB(X):={T ⊂ χ: every additive functional onX, bounded from above onT, is continuous} are equal. This generalizes a result of Marcin E. Kuczma (1970) who has shown the equalityA(ℝn)=B(ℝn) However, the infinite dimensional case requires completely different methods; therefore, even in the caseX = ℝ n we obtain a new (and perhaps simpler) proof than that given by M. E. Kuczma.
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Ger, R., Kominek, Z. Boundedness and continuity of additive and convex functionals. Aeq. Math. 37, 252–258 (1989). https://doi.org/10.1007/BF01836447
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DOI: https://doi.org/10.1007/BF01836447