Extension of Simons’ inequality
HTML articles powered by AMS MathViewer
- by Kersti Kivisoo and Eve Oja PDF
- Proc. Amer. Math. Soc. 133 (2005), 3485-3496 Request permission
Abstract:
We prove the following extended version of Simons’ inequality and present its applications. Let $S$ be a set and $T$ be a subset of $S$. Let $C$ be a subset of a Hausdorff topological vector space which is invariant under infinite convex combinations. Let $f: C\times S \longrightarrow \mathbb {R}$ be a bounded function such that the functions $f( \cdot , t):C\longrightarrow \mathbb {R}$ are convex for all $t \in T$ and $f(\lambda x, s)=\lambda f(x, s)$ whenever $\lambda >0$, $x, \lambda x \in C$ and $s\in S.$ Let $(x_n)$ be a sequence in $C$. Assume that, for every $x \in C_1 =\left \{\sum _{n=1}^{\infty }\lambda _n x_n :\quad \lambda _n\geq 0, \sum _{n=1}^{\infty }\lambda _n=1 \right \}$, there exists $t \in T$ satisfying $f(x, t)=\sup _{s\in S} f(x, s)$. Then \[ \inf _{x\in C_1}\sup _{s\in S}f(x, s) \leq \sup _{t\in T}\limsup _{n}f(x_n, t).\] If $-C_1\subset C$, then the set $C_1$ in the above inequality can be replaced by $\textrm {conv}\{x_1, x_2, \ldots \}$.References
- María D. Acosta and Manuel Ruiz Galán, New characterizations of the reflexivity in terms of the set of norm attaining functionals, Canad. Math. Bull. 41 (1998), no. 3, 279–289. MR 1637649, DOI 10.4153/CMB-1998-040-x
- Robert Deville and Catherine Finet, An extension of Simons’ inequality and applications, Rev. Mat. Complut. 14 (2001), no. 1, 95–104. MR 1851724, DOI 10.5209/rev_{R}EMA.2001.v14.n1.17044
- Marián Fabián and Gilles Godefroy, The dual of every Asplund space admits a projectional resolution of the identity, Studia Math. 91 (1988), no. 2, 141–151. MR 985081, DOI 10.4064/sm-91-2-141-151
- Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, and Václav Zizler, Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8, Springer-Verlag, New York, 2001. MR 1831176, DOI 10.1007/978-1-4757-3480-5
- M. Ruiz Galán and S. Simons, A new minimax theorem and a perturbed James’s theorem, Bull. Austral. Math. Soc. 66 (2002), no. 1, 43–56. MR 1922606, DOI 10.1017/S0004972700020669
- Gilles Godefroy, Boundaries of a convex set and interpolation sets, Math. Ann. 277 (1987), no. 2, 173–184. MR 886417, DOI 10.1007/BF01457357
- Gilles Godefroy, Some applications of Simons’ inequality, Serdica Math. J. 26 (2000), no. 1, 59–78. MR 1767034
- Gilles Godefroy and Václav Zizler, Roughness properties of norms on non-Asplund spaces, Michigan Math. J. 38 (1991), no. 3, 461–466. MR 1116501, DOI 10.1307/mmj/1029004394
- P. Habala, P. Hájek, V. Zizler, Introduction to Banach Spaces, I, Charles University, Prague, 1996.
- Åsvald Lima, Property $(wM^\ast )$ and the unconditional metric compact approximation property, Studia Math. 113 (1995), no. 3, 249–263. MR 1330210, DOI 10.4064/sm-113-3-249-263
- Eve Oja, A proof of the Simons inequality, Acta Comment. Univ. Tartu. Math. 2 (1998), 27–28. MR 1714730
- Eve Oja, Géométrie des espaces de Banach ayant des approximations de l’identité contractantes, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 12, 1167–1170 (French, with English and French summaries). MR 1701379, DOI 10.1016/S0764-4442(99)80433-9
- Eve Oja, Geometry of Banach spaces having shrinking approximations of the identity, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2801–2823. MR 1675226, DOI 10.1090/S0002-9947-00-02521-6
- S. Simons, A convergence theorem with boundary, Pacific J. Math. 40 (1972), 703–708. MR 312193, DOI 10.2140/pjm.1972.40.703
- Stephen Simons, An eigenvector proof of Fatou’s lemma for continuous functions, Math. Intelligencer 17 (1995), no. 3, 67–70. MR 1347898, DOI 10.1007/BF03024373
Additional Information
- Kersti Kivisoo
- Affiliation: Faculty of Mathematics and Computer Science, Tartu University, J. Liivi 2, EE-50409 Tartu, Estonia
- Email: kersti.kivisoo@mail.ee
- Eve Oja
- Affiliation: Faculty of Mathematics and Computer Science, Tartu University, J. Liivi 2, EE-50409 Tartu, Estonia
- Email: eveoja@math.ut.ee
- Received by editor(s): July 2, 2004
- Published electronically: June 28, 2005
- Additional Notes: This research was partially supported by Estonian Science Foundation Grant 5704
- Communicated by: Jonathan M. Borwein
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 3485-3496
- MSC (2000): Primary 39B62, 46A55, 46B20, 54C30
- DOI: https://doi.org/10.1090/S0002-9939-05-08267-5
- MathSciNet review: 2163583