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

Abstract

top
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.

How to cite

top

Godefroy, 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
  1. Contreras M.D., Payá R., On upper semicontinuity of duality mapping, Proc. Amer. Math. Soc. 121 (1994), 451-459. (1994) MR1215199
  2. 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
  3. 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
  4. Franchetti C., Lipschitz maps and the geometry of the unit ball in normed spaces, Archiv. Math. 46 (1986), 76-84. (1986) Zbl0564.46014MR0829819
  5. Franchetti C., Payá R., Banach spaces with strongly subdifferentiable norm, Boll. Uni. Mat. Ital. VII-B (1993), 45-70. (1993) MR1216708
  6. Godefroy G., Some applications of Simons' inequality, Seminar of Functional Analysis II, Univ. of Murcia, to appear. MR1767034
  7. Godefroy G., Kalton N.J., The ball topology and its applications, Contemporary Math. 85 (1989), 195-238. (1989) Zbl0676.46003MR0983386
  8. Gregory D.A., Upper semicontinuity of subdifferential mappings, Canad. Math. Bull. 23 (1980), 11-19. (1980) MR0573553
  9. Haydon R., Trees in renorming theory, to appear. Zbl1036.46003MR1674838
  10. James R.C., Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129-140. (1964) Zbl0129.07901MR0165344
  11. Jayne J.E., Rogers C.A., Borel selectors for upper semicontinuous set valued mappings, Acta Math. 155 (1985), 41-79. (1985) MR0793237
  12. John K., Zizler V., Smoothness and its equivalents in weakly compactly generated Banach spaces, J. Funct. Anal. 15 (1974), 161-166. (1974) Zbl0272.46012MR0417759
  13. John K., Zizler V., Projections in dual weakly compactly generated Banach spaces, Studia Math. 49 (1973), 41-50. (1973) Zbl0247.46029MR0336295
  14. Lindenstrauss J., Tzafriri L., Classical Banach spaces I, Sequence Spaces Springer-Verlag (1977). (1977) Zbl0362.46013MR0500056
  15. Odell E., Rosenthal H.P., A double dual characterization of separable Banach spaces containing 1 , Israel J. Math. 20 (1975), 375-384. (1975) Zbl0312.46031MR0377482
  16. Simons S., A convergence theorem with boundary, Pacific J. Math. 40 (1972), 703-708. (1972) Zbl0237.46012MR0312193
  17. Singer I., Bases in Banach Spaces II, Springer-Verlag (1981). (1981) Zbl0467.46020MR0610799
  18. Stegall C., The Radon-Nikodym property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 (1975), 213-223. (1975) Zbl0318.46056MR0374381
  19. Troyanski S.L., On a property of the norm which is close to local uniform convexity, Math. Ann. 27 (1985), 305-313. (1985) MR0783556

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.