Additivity, subadditivity and linearity: Automatic continuity and quantifier weakening

We study the interplay between additivity (as in the Cauchy functional equation), subadditivity and linearity. We obtain automatic continuity results in which additive or subadditive functions, under minimal regularity conditions, are continuous and so linear. We apply our results in the context of...

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Bibliographic Details
Published in:Indagationes mathematicae 2018-04, Vol.29 (2), p.687-713
Main Authors: Bingham, N.H., Ostaszewski, A.J.
Format: Article
Language:English
Subjects:
Additive subgroup
Analytic spanning set
Functional equations
Hamel basis
Heiberg–Seneta conditions
Linearity
Mathematics
Proof theory
Regular variation
Shift-compact
Steinhaus sum theorem
Subadditive
Sublinear
Thinning
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Summary:We study the interplay between additivity (as in the Cauchy functional equation), subadditivity and linearity. We obtain automatic continuity results in which additive or subadditive functions, under minimal regularity conditions, are continuous and so linear. We apply our results in the context of quantifier weakening in the theory of regular variation, completing our programme of reducing the number of hard proofs there to zero.
ISSN:0019-3577
1872-6100
DOI:10.1016/j.indag.2017.11.005