Almost linear operators and functionals on $\mathcal {C}([0,1])$
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- by J. R. Baxter and R. V. Chacon PDF
- Proc. Amer. Math. Soc. 47 (1975), 147-154 Request permission
Abstract:
Let $\mathcal {C}(M)$ be the bounded continuous functions on a topological space $M$. โAlmost linearโ 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) $||f|| \to 0$ 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.References
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Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 47 (1975), 147-154
- MSC: Primary 28A10
- DOI: https://doi.org/10.1090/S0002-9939-1975-0352380-9
- MathSciNet review: 0352380