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Keywords:
James Theorem; Bishop-Phelps Theorem; smooth variational principles
James Theorem; Bishop-Phelps Theorem; smooth variational principles
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.
References:
[1] Azagra D., Deville R.: James’ theorem fails for starlike bodies. J. Funct. Anal. 180 (2) (2001), 328–346. MR 1814992 | Zbl 0983.46016
[2] Bishop E., Phelps R. R.: The support cones in Banach spaces and their applications. Adv. Math. 13 (1974), 1–19. MR 0338741
[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
[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. MR 1200641 | Zbl 0774.49021
[5] Deville R., Maaden A.: Smooth variational principles in Radon-Nikodým spaces. Bull. Austral. Math. Soc. 60 (1999), 109–118. MR 1702818 | Zbl 0956.49003
[6] Diestel J.: Geometry of Banach spaces - Selected topics. Lecture Notes in Math., Berlin – Heidelberg – New York 485 (1975). MR 0461094 | Zbl 0307.46009
[7] Fabian M., Mordukhovich B. S.: Separable reduction and extremal principles in variational analysis. Nonlinear Anal. 49 (2) (2002), 265–292. MR 1885121 | Zbl 1061.49015
[8] Haydon R.: A counterexample in several question about scattered compact spaces. Bull. London Math. Soc. 22 (1990), 261–268. MR 1041141
[9] Ioffe A. D.: Proximal analysis and approximate subdifferentials. J. London Math. Soc. (2) 41 (1990), 175–192. MR 1063554 | Zbl 0725.46045
[10] James R. C.: Weakly compact sets. Trans. Amer. Math. Soc. 113 (1964), 129–140. MR 0165344 | Zbl 0129.07901
[11] Leduc M.: Densité de certaines familles d’hyperplans tangents. C. R. Acad. Sci. Paris, Sér. A 270 (1970), 326–328. MR 0276733 | Zbl 0193.10801
[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
[13] Mordukhovich B. S., Shao Y. H.: Extremal characterizations of Asplund spaces. Proc. Amer. Math. Soc. 124 (1) (1996), 197–205. MR 1291788 | Zbl 0849.46010
[14] Mordukhovich B. S., Shao Y. H.: Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc. 348 (4) (1996), 1235–1280. MR 1333396 | Zbl 0881.49009
[15] Phelps R. R.: Convex functions, Monotone Operators and Differentiability. Lecture Notes in Math., Berlin – Heidelberg – New York – London – Paris – Tokyo 1364 (1991).
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