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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Almost linear operators and functionals on $ \mathcal{C}([0,1])$


Authors: J. R. Baxter and R. V. Chacon
Journal: Proc. Amer. Math. Soc. 47 (1975), 147-154
MSC: Primary 28A10
MathSciNet review: 0352380
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Abstract: Let $ \mathcal{C}(M)$ be the bounded continuous functions on a topological space $ M$. ``Almost linear'' operators (and functionals) on $ \mathcal{C}(M)$ are defined. Almost linearity does not imply linearity in general. However, it is shown that if $ M = [0,1]$ then any almost linear operator (or functional) must be linear. Specifically, if (a) $ \vert\vert f\vert\vert \to 0$ implies $ T(f) \to 0$, (b) $ T(f + g) = T(f) + T(g)$ whenever $ fg = 0$, (c) $ T(f + g) = T(f) + T(g)$ whenever $ g$ is constant, and $ M = [0,1]$, then $ T$ is linear. An application is given to convergence of measures.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0352380-9
PII: S 0002-9939(1975)0352380-9
Article copyright: © Copyright 1975 American Mathematical Society