2018

DOI: 10.1007/s00010-018-0618-8

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Variants on the Berz sublinearity theorem

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Adam Ostaszewski

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Abstract: To Roy O. Davies on his 90 th birthday.Abstract. We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function -results of quantifierweakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature.

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“…Note that HB for separable normed spaces is not provable from DC [DodM,Cor. 4], unless the space is complete -see [BinO10].…”

Section: Introductionmentioning

confidence: 99%

Set theory and the analyst

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²

2018

European Journal of Mathematics

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Then to the rolling heaven itself I cried, Asking what lamp had destiny to guide Her little children stumbling in the dark. And 'A blind understanding' heaven replied.

“…Note that HB for separable normed spaces is not provable from DC [DodM,Cor. 4], unless the space is complete -see [BinO10].…”

Section: Introductionmentioning

confidence: 99%

Set theory and the analyst

¹

,

²

2018

European Journal of Mathematics

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Then to the rolling heaven itself I cried, Asking what lamp had destiny to guide Her little children stumbling in the dark. And 'A blind understanding' heaven replied.

Further remarks on local K-boundedness of K-subadditive set-valued maps

Jabłońska,

Nikodem

2023

Aequat. Math.

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Let X be an abelian metric group with an invariant metric, Y be a real normed space and K be a convex cone in Y. We prove that a K-subadditive (K-superadditive) compact- and convex-valued map $$F:X\rightarrow \mathcal{C}\mathcal{C}(Y)$$ F : X → C C ( Y ) , for which the functionals $$f_{y^*}(x)=\inf y^*(F(x))$$ f y ∗ ( x ) = inf y ∗ ( F ( x ) ) are lower (upper, resp.) semicontinuous for any real continuous and non-negative on K functional $$y^*$$ y ∗ , has to be locally K-bounded on X. Our results refer to the papers Banakh and Jabłońska (Israel J Math 230:361–386, 2019), Jabłońska and Nikodem (Math Inequal Appl 22:1081–1089, 2019) and Nikodem (Aequationes Math 62:175–183, 2001).

On Subadditive Functions Bounded Above on a “Large” Set

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et al. 2020

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It is well known that boundedness of a subadditive function need not imply its continuity. Here we prove that each subadditive function f : X → R bounded above on a shift-compact (non-Haar-null, non-Haar-meagre) set is locally bounded at each point of the domain. Our results refer to [31, Chapter 16] and papers

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