Article

Full entry | PDF   (0.2 MB)
Keywords:
Fréchet differentiability; convex functions; Asplund spaces
Summary:
A class of convex functions where the sets of subdifferentials behave like the unit ball of the dual space of an Asplund space is found. These functions, which we called Asplund functions also possess some stability properties. We also give a sufficient condition for a function to be an Asplund function in terms of the upper-semicontinuity of the subdifferential map.
References:
[C-P] Contreras M.D., Payá R.: On upper semicontinuity of duality mappings. Proc. Amer. Math. Soc. 112 (1994), 451-459. MR 1215199
[D-G-Z] Deville R., Godefroy G., Zizler V.: Smoothness and Renormings in Banach Spaces. Pitman Monograph and Survey in Pure and Applied Mathematics \bf{64}. MR 1211634 | Zbl 0782.46019
[Du-N] Dulst D.V., Namioka I.: A note on trees in conjugate Banach spaces. Indag. Math. 46 (1984), 7-10. MR 0748973 | Zbl 0537.46025
[N-Ph] Namioka I., Phelps R.R.: Banach spaces which are Asplund spaces. Duke Math J. 42 (1975), 735-749. MR 0390721 | Zbl 0332.46013
[F] Fabian M.: On projectional resolution of identity on the duals of certain Banach spaces. Bull. Austral. Math. Soc. 35 (1987), 363-371. MR 0888895
[Go] Godefroy G.: Some applications of Simons' inequality. Sem. Funct. Anal., University of Murcia, to appear. MR 1767034
[Gi] Giles J.R.: On the characterisation of Asplund spaces. J. Austral. Math. Soc. (Ser. A) 32 (1982), 134-144. MR 0643437 | Zbl 0486.46019
[Gi-Gr-Si] Giles J.R., Gregory D.A., Sims B.: Geometric implications of upper semi-continuity of the duality mapping on a Banach space. Pacific J. Math. 79 (1978), 99-109. MR 0526669
[Gi-Sc] Giles J.R., Sciffer S.: Separable determination of Fréchet differentiability of convex functions. Bull. Austral. Math. Soc. 52 (1995), 161-167. MR 1344269 | Zbl 0839.46036
[J-N-R] Jayne J.E., Namioka I., Rogers C.A.: $\sigma$-fragmentable Banach spaces. Mathematika 39 (1992), 161-188. MR 1176478 | Zbl 0761.46009
[Ph] Phelps R.R.: Convex Functions, Monotone Operators and Differentiability. Lect. Notes. in Math., Springer-Verlag 1364 (1993) (Second Edition). MR 1238715 | Zbl 0921.46039
[Pr] Preiss D.: Gâteaux differentiable functions are somewhere Fréchet differentiable. Rend. Cir. Mat. Pal. 33 (1984), 122-133. MR 0743214 | Zbl 0573.46024
[Pr-Z] Preiss D., Zajíček D.: Fréchet differentiability of convex functions in Banach space with separable duals. Proc. Amer. Math. Soc. 91 (1984), 202-204. MR 0740171
[S] Simons S.: A convergence theorem with boundary. Pacific J. Math. 40 (1972), 703-708. MR 0312193 | Zbl 0237.46012
[St] Stegall C.: The Radon-Nikodym property in conjugate Banach spaces. Trans. Amer. Math. Soc. 206 (1975), 213-223. MR 0374381 | Zbl 0318.46056
[T$_{1}$] Tang W.-K.: On Fréchet differentiability of convex functions on Banach spaces. Comment. Math. Univ. Carolinae 36 (1995), 249-253. MR 1357526 | Zbl 0831.46045
[T$_{2}$] Tang W.-K.: Sets of differentials and smoothness of convex functions. Bull. Austral. Math. Soc. 52 (1995), 91-96. MR 1344263 | Zbl 0839.46008
[Y] Yost D.: Asplund spaces for beginners. Acta Univ. Carolinae 34 (1993), 159-177. MR 1282979 | Zbl 0815.46022

Partner of