Some properties on the closed subsets in Banach spaces

Abdelhakim Maaden; Abdelkader Stouti

Archivum Mathematicum (2006)

  • Volume: 042, Issue: 2, page 167-174
  • ISSN: 0044-8753

Abstract

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It is shown that under natural assumptions, there exists a linear functional does not have supremum on a closed bounded subset. That is the James Theorem for non-convex bodies. Also, a non-linear version of the Bishop-Phelps Theorem and a geometrical version of the formula of the subdifferential of the sum of two functions are obtained.

How to cite

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Maaden, Abdelhakim, and Stouti, Abdelkader. "Some properties on the closed subsets in Banach spaces." Archivum Mathematicum 042.2 (2006): 167-174. <http://eudml.org/doc/249825>.

@article{Maaden2006,
abstract = {It is shown that under natural assumptions, there exists a linear functional does not have supremum on a closed bounded subset. That is the James Theorem for non-convex bodies. Also, a non-linear version of the Bishop-Phelps Theorem and a geometrical version of the formula of the subdifferential of the sum of two functions are obtained.},
author = {Maaden, Abdelhakim, Stouti, Abdelkader},
journal = {Archivum Mathematicum},
keywords = {James Theorem; Bishop-Phelps Theorem; smooth variational principles; James theorem; Bishop-Phelps theorem; smooth variational principles},
language = {eng},
number = {2},
pages = {167-174},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Some properties on the closed subsets in Banach spaces},
url = {http://eudml.org/doc/249825},
volume = {042},
year = {2006},
}

TY - JOUR
AU - Maaden, Abdelhakim
AU - Stouti, Abdelkader
TI - Some properties on the closed subsets in Banach spaces
JO - Archivum Mathematicum
PY - 2006
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 042
IS - 2
SP - 167
EP - 174
AB - It is shown that under natural assumptions, there exists a linear functional does not have supremum on a closed bounded subset. That is the James Theorem for non-convex bodies. Also, a non-linear version of the Bishop-Phelps Theorem and a geometrical version of the formula of the subdifferential of the sum of two functions are obtained.
LA - eng
KW - James Theorem; Bishop-Phelps Theorem; smooth variational principles; James theorem; Bishop-Phelps theorem; smooth variational principles
UR - http://eudml.org/doc/249825
ER -

References

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  10. James R. C., Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129–140. (1964) Zbl0129.07901MR0165344
  11. Leduc M., Densité de certaines familles d’hyperplans tangents, C. R. Acad. Sci. Paris, Sér. A 270 (1970), 326–328. (1970) Zbl0193.10801MR0276733
  12. Mordukhovich B. S., Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Sov. Math. Dokl. 22 (1980), 526–530. (1980) Zbl0491.49011
  13. Mordukhovich B. S., Shao Y. H., Extremal characterizations of Asplund spaces, Proc. Amer. Math. Soc. 124 (1) (1996), 197–205. (1996) Zbl0849.46010MR1291788
  14. Mordukhovich B. S., Shao Y. H., Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348 (4) (1996), 1235–1280. (1996) Zbl0881.49009MR1333396
  15. Phelps R. R., Convex functions, Monotone Operators and Differentiability, Lecture Notes in Math., Berlin – Heidelberg – New York – London – Paris – Tokyo 1364 (1991). (1991) 

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