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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Riesz seminorms with Fatou properties


Author: C. D. Aliprantis
Journal: Proc. Amer. Math. Soc. 45 (1974), 383-388
MSC: Primary 46A40
DOI: https://doi.org/10.1090/S0002-9939-1974-0350371-4
MathSciNet review: 0350371
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Abstract: A seminormed Riesz space $ {L_\rho }$ satisfies the $ \sigma $-Fatou property (resp. the Fatou property) if $ \theta \leq {u_n} \uparrow u$ in $ L$ (resp. $ \theta \leq {u_\alpha } \uparrow u\;{\text{in}}\;L$) implies $ \rho ({u_n}) \uparrow \rho (u)$ (resp. $ \rho ({u_\alpha }) \uparrow \rho (u)$). The following results are proved:

(i) A normed Riesz space $ {L_\rho }$ satisfies the $ \sigma $-Fatou property if, and only if, its norm completion does and $ {L_\rho }$ has $ ({\mathbf{A}},0)$.

(ii) The quotient space $ {L_\rho }/{I_\rho }$ has the Fatou property if $ {L_\rho }$ is Archimedean with the Fatou property. $ ({I_\rho } = \{ u\varepsilon L:\rho (u) = 0\} .)$

(iii) If $ {L_\rho }$ is almost $ \sigma $-Dedekind complete with the $ \sigma $-Fatou property, then $ {L_\rho }/{I_\rho }$ has the $ \sigma $-Fatou property.

A counterexample shows that (iii) may be false for Archimedean Riesz spaces.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0350371-4
Keywords: Riesz spaces, $ \sigma $-Fatou property, norm completion, quotient Riesz spaces
Article copyright: © Copyright 1974 American Mathematical Society