# Strong subdifferentiability of norms and geometry of Banach spaces

Gilles Godefroy; Vicente Montesinos; Václav Zizler

Commentationes Mathematicae Universitatis Carolinae (1995)

- Volume: 36, Issue: 3, page 493-502
- ISSN: 0010-2628

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topGodefroy, Gilles, Montesinos, Vicente, and Zizler, Václav. "Strong subdifferentiability of norms and geometry of Banach spaces." Commentationes Mathematicae Universitatis Carolinae 36.3 (1995): 493-502. <http://eudml.org/doc/247756>.

@article{Godefroy1995,

abstract = {The strong subdifferentiability of norms (i.eȯne-sided differentiability uniform in directions) is studied in connection with some structural properties of Banach spaces. It is shown that every separable Banach space with nonseparable dual admits a norm that is nowhere strongly subdifferentiable except at the origin. On the other hand, every Banach space with a strongly subdifferentiable norm is Asplund.},

author = {Godefroy, Gilles, Montesinos, Vicente, Zizler, Václav},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {strong subdifferentiability of norms; Asplund spaces; renormings; weak compact generating; Asplund spaces; renormings; weak compact generating; strong subdifferentiability of norms; structural properties of Banach spaces; separable Banach space with nonseparable dual},

language = {eng},

number = {3},

pages = {493-502},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Strong subdifferentiability of norms and geometry of Banach spaces},

url = {http://eudml.org/doc/247756},

volume = {36},

year = {1995},

}

TY - JOUR

AU - Godefroy, Gilles

AU - Montesinos, Vicente

AU - Zizler, Václav

TI - Strong subdifferentiability of norms and geometry of Banach spaces

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 1995

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 36

IS - 3

SP - 493

EP - 502

AB - The strong subdifferentiability of norms (i.eȯne-sided differentiability uniform in directions) is studied in connection with some structural properties of Banach spaces. It is shown that every separable Banach space with nonseparable dual admits a norm that is nowhere strongly subdifferentiable except at the origin. On the other hand, every Banach space with a strongly subdifferentiable norm is Asplund.

LA - eng

KW - strong subdifferentiability of norms; Asplund spaces; renormings; weak compact generating; Asplund spaces; renormings; weak compact generating; strong subdifferentiability of norms; structural properties of Banach spaces; separable Banach space with nonseparable dual

UR - http://eudml.org/doc/247756

ER -

## References

top- Contreras M.D., Payá R., On upper semicontinuity of duality mapping, Proc. Amer. Math. Soc. 121 (1994), 451-459. (1994) MR1215199
- Deville R., Godefroy G., Zizler V., Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64 (1993). (1993) Zbl0782.46019MR1211634
- Deville R., Godefroy G., Hare D., Zizler V., Differentiability of convex functions and the convex point of continuity property in Banach spaces, Israel J. Math. 59 (1987), 245-255. (1987) Zbl0654.46021MR0920087
- Franchetti C., Lipschitz maps and the geometry of the unit ball in normed spaces, Archiv. Math. 46 (1986), 76-84. (1986) Zbl0564.46014MR0829819
- Franchetti C., Payá R., Banach spaces with strongly subdifferentiable norm, Boll. Uni. Mat. Ital. VII-B (1993), 45-70. (1993) MR1216708
- Godefroy G., Some applications of Simons' inequality, Seminar of Functional Analysis II, Univ. of Murcia, to appear. MR1767034
- Godefroy G., Kalton N.J., The ball topology and its applications, Contemporary Math. 85 (1989), 195-238. (1989) Zbl0676.46003MR0983386
- Gregory D.A., Upper semicontinuity of subdifferential mappings, Canad. Math. Bull. 23 (1980), 11-19. (1980) MR0573553
- Haydon R., Trees in renorming theory, to appear. Zbl1036.46003MR1674838
- James R.C., Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129-140. (1964) Zbl0129.07901MR0165344
- Jayne J.E., Rogers C.A., Borel selectors for upper semicontinuous set valued mappings, Acta Math. 155 (1985), 41-79. (1985) MR0793237
- John K., Zizler V., Smoothness and its equivalents in weakly compactly generated Banach spaces, J. Funct. Anal. 15 (1974), 161-166. (1974) Zbl0272.46012MR0417759
- John K., Zizler V., Projections in dual weakly compactly generated Banach spaces, Studia Math. 49 (1973), 41-50. (1973) Zbl0247.46029MR0336295
- Lindenstrauss J., Tzafriri L., Classical Banach spaces I, Sequence Spaces Springer-Verlag (1977). (1977) Zbl0362.46013MR0500056
- Odell E., Rosenthal H.P., A double dual characterization of separable Banach spaces containing ${\ell}_{1}$, Israel J. Math. 20 (1975), 375-384. (1975) Zbl0312.46031MR0377482
- Simons S., A convergence theorem with boundary, Pacific J. Math. 40 (1972), 703-708. (1972) Zbl0237.46012MR0312193
- Singer I., Bases in Banach Spaces II, Springer-Verlag (1981). (1981) Zbl0467.46020MR0610799
- Stegall C., The Radon-Nikodym property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 (1975), 213-223. (1975) Zbl0318.46056MR0374381
- Troyanski S.L., On a property of the norm which is close to local uniform convexity, Math. Ann. 27 (1985), 305-313. (1985) MR0783556

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