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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The weak stability of the positive face in $L^ 1$
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by Zhibao Hu PDF
Proc. Amer. Math. Soc. 123 (1995), 131-134 Request permission

Abstract:

Let F be the positive face of the unit ball of ${L^1}[0,1]$. We show that F is weakly stable in the sense that the midpoint map ${\Phi _{1/2}}:F \times F \to F$, with ${\Phi _{1/2}}(f,g) = \frac {1}{2}(f + g)$, is open with respect to the weak topology. This weak stability of the set F is the reason behind the fact that the notions of "huskable" and "strongly regular" operators coincide for operators from ${L^1}[0,1]$ to a Banach space X. We prove this stability by showing that if ${f_1},{f_2} \in F,\lambda \in (0,1),\varepsilon > 0$ and $\delta \geq \max \{ 2\varepsilon /\lambda ,2\varepsilon /(1 - \lambda )\}$, then \[ \lambda {V_{P,\delta }}({f_1}) + (1 - \lambda ){V_{P,\delta }}({f_2}) \supset {V_{P,\varepsilon }}[\lambda {f_1} + (1 - \lambda ){f_2}],\] where $P = \{ {A_1}, \ldots ,{A_n}\}$ is a finite positive partition of [0, 1] and \[ {V_{P,\varepsilon }}(f) = \left \{ {g \in F:\sum \limits _{i = 1}^n {\left | {\int _{{A_i}} {(f - g)(t)d\mu (t)} } \right | \leq \varepsilon } } \right \}\] for any f in F. We construct an example showing that for any $0 < \lambda < 1$ there are functions ${f_1}$ and ${f_2}$ in F such that if $0 < \varepsilon < 2\min \{ \lambda ,1 - \lambda \}$ and $0 \leq \delta < \max \{ \varepsilon /\lambda ,\varepsilon /(1 - \lambda )\}$, then \[ \lambda {V_{P,\delta }}({f_1}) + (1 - \lambda ){V_{P,\delta }}({f_2})\not \supset {V_{P,\varepsilon }}(\lambda {f_1} + (1 - \lambda ){f_2}).\] Thus the "formula" that $\lambda {V_{p,\varepsilon }}({f_1}) + (1 - \lambda ){V_{p,\varepsilon }}({f_2}) = {V_{p,\varepsilon }}(\lambda {f_1} + (1 - \lambda ){f_2})$ given by Ghoussoub et al. in Mem. Amer. Math. Soc., vol. 70, no. 378, which is used there to establish the weak stability of F, is false.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 131-134
  • MSC: Primary 46E30; Secondary 46B20
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1213862-3
  • MathSciNet review: 1213862