| Title:
|
Some properties on the closed subsets in Banach spaces (English) |
| Author:
|
Maaden, Abdelhakim |
| Author:
|
Stouti, Abdelkader |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
42 |
| Issue:
|
2 |
| Year:
|
2006 |
| Pages:
|
167-174 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
James Theorem |
| Keyword:
|
Bishop-Phelps Theorem |
| Keyword:
|
smooth variational principles |
| MSC:
|
46B20 |
| MSC:
|
49J52 |
| idZBL:
|
Zbl 1164.46307 |
| idMR:
|
MR2240354 |
| . |
| Date available:
|
2008-06-06T22:47:50Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107993 |
| . |
| Reference:
|
[1] Azagra D., Deville R.: James’ theorem fails for starlike bodies.J. Funct. Anal. 180 (2) (2001), 328–346. Zbl 0983.46016, MR 1814992 |
| Reference:
|
[2] Bishop E., Phelps R. R.: The support cones in Banach spaces and their applications.Adv. Math. 13 (1974), 1–19. MR 0338741 |
| Reference:
|
[3] Deville R., El Haddad E.: The subdifferential of the sum of two functions in Banach spaces, I. First order case.J. Convex Anal. 3 (2) (1996), 295–308. MR 1448058 |
| Reference:
|
[4] Deville R., Godefroy G., Zizler V.: A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions.J. Funct. Anal. 111 (1993), 197–212. Zbl 0774.49021, MR 1200641 |
| Reference:
|
[5] Deville R., Maaden A.: Smooth variational principles in Radon-Nikodým spaces.Bull. Austral. Math. Soc. 60 (1999), 109–118. Zbl 0956.49003, MR 1702818 |
| Reference:
|
[6] Diestel J.: Geometry of Banach spaces - Selected topics.Lecture Notes in Math., Berlin – Heidelberg – New York 485 (1975). Zbl 0307.46009, MR 0461094 |
| Reference:
|
[7] Fabian M., Mordukhovich B. S.: Separable reduction and extremal principles in variational analysis.Nonlinear Anal. 49 (2) (2002), 265–292. Zbl 1061.49015, MR 1885121 |
| Reference:
|
[8] Haydon R.: A counterexample in several question about scattered compact spaces.Bull. London Math. Soc. 22 (1990), 261–268. MR 1041141 |
| Reference:
|
[9] Ioffe A. D.: Proximal analysis and approximate subdifferentials.J. London Math. Soc. (2) 41 (1990), 175–192. Zbl 0725.46045, MR 1063554 |
| Reference:
|
[10] James R. C.: Weakly compact sets.Trans. Amer. Math. Soc. 113 (1964), 129–140. Zbl 0129.07901, MR 0165344 |
| Reference:
|
[11] Leduc M.: Densité de certaines familles d’hyperplans tangents.C. R. Acad. Sci. Paris, Sér. A 270 (1970), 326–328. Zbl 0193.10801, MR 0276733 |
| Reference:
|
[12] Mordukhovich B. S.: Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems.Sov. Math. Dokl. 22 (1980), 526–530. Zbl 0491.49011 |
| Reference:
|
[13] Mordukhovich B. S., Shao Y. H.: Extremal characterizations of Asplund spaces.Proc. Amer. Math. Soc. 124 (1) (1996), 197–205. Zbl 0849.46010, MR 1291788 |
| Reference:
|
[14] Mordukhovich B. S., Shao Y. H.: Nonsmooth sequential analysis in Asplund spaces.Trans. Amer. Math. Soc. 348 (4) (1996), 1235–1280. Zbl 0881.49009, MR 1333396 |
| Reference:
|
[15] Phelps R. R.: Convex functions, Monotone Operators and Differentiability.Lecture Notes in Math., Berlin – Heidelberg – New York – London – Paris – Tokyo 1364 (1991). |
| . |
