On Lipschitz and d.c. surfaces of finite codimension in a Banach space

Luděk Zajíček

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 3, page 849-864
  • ISSN: 0011-4642

Abstract

top
Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space and properties of generated σ -ideals are studied. These σ -ideals naturally appear in the differentiation theory and in the abstract approximation theory. Using these properties, we improve an unpublished result of M. Heisler which gives an alternative proof of a result of D. Preiss on singular points of convex functions.

How to cite

top

Zajíček, Luděk. "On Lipschitz and d.c. surfaces of finite codimension in a Banach space." Czechoslovak Mathematical Journal 58.3 (2008): 849-864. <http://eudml.org/doc/37872>.

@article{Zajíček2008,
abstract = {Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space and properties of generated $\sigma $-ideals are studied. These $\sigma $-ideals naturally appear in the differentiation theory and in the abstract approximation theory. Using these properties, we improve an unpublished result of M. Heisler which gives an alternative proof of a result of D. Preiss on singular points of convex functions.},
author = {Zajíček, Luděk},
journal = {Czechoslovak Mathematical Journal},
keywords = {Banach space; Lipschitz surface; d.c. surface; multiplicity points of monotone operators; singular points of convex functions; Aronszajn null sets; multiplicity points of monotone operators; singular points of convex functions; Aronszajn null sets},
language = {eng},
number = {3},
pages = {849-864},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Lipschitz and d.c. surfaces of finite codimension in a Banach space},
url = {http://eudml.org/doc/37872},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Zajíček, Luděk
TI - On Lipschitz and d.c. surfaces of finite codimension in a Banach space
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 3
SP - 849
EP - 864
AB - Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space and properties of generated $\sigma $-ideals are studied. These $\sigma $-ideals naturally appear in the differentiation theory and in the abstract approximation theory. Using these properties, we improve an unpublished result of M. Heisler which gives an alternative proof of a result of D. Preiss on singular points of convex functions.
LA - eng
KW - Banach space; Lipschitz surface; d.c. surface; multiplicity points of monotone operators; singular points of convex functions; Aronszajn null sets; multiplicity points of monotone operators; singular points of convex functions; Aronszajn null sets
UR - http://eudml.org/doc/37872
ER -

References

top
  1. Berkson, B., 10.2140/pjm.1963.13.7, Pacific J. Math. 13 (1963), 7-22. (1963) MR0152869DOI10.2140/pjm.1963.13.7
  2. Benyamini, Y., Lindenstrauss, J., Geometric Nonlinear Functional Analysis, Vol. 1, Colloqium publications (American Mathematical Society); v. 48, Providence, Rhode Island (2000). (2000) MR1727673
  3. Duda, J., On inverses of δ -convex mappings, Comment. Math. Univ. Carolin. 42 (2001), 281-297. (2001) Zbl1053.47522MR1832147
  4. Erdös, P., 10.1090/S0002-9904-1946-08514-6, Bull. Amer. Math. Soc. 52 (1946), 107-109. (1946) MR0015144DOI10.1090/S0002-9904-1946-08514-6
  5. Gohberg, I. C., Krein, M. G., Fundamental aspects of defect numbers, root numbers, and indexes of linear operators, Uspekhi Mat. Nauk 12 (1957), 43-118 Russian. (1957) MR0096978
  6. Hartman, P., 10.2140/pjm.1959.9.707, Pacific J. Math. 9 (1959), 707-713. (1959) Zbl0093.06401MR0110773DOI10.2140/pjm.1959.9.707
  7. Heisler, M., Some aspects of differentiability in geometry on Banach spaces, Ph.D. thesis, Charles University, Prague (1996). (1996) 
  8. Kato, T., Perturbation Theory for Linear Operators, Springer-Verlag, Berin (1976). (1976) Zbl0342.47009MR0407617
  9. Kopecká, E., Malý, J., Remarks on delta-convex functions, Comment. Math. Univ. Carolin. 31 (1990), 501-510. (1990) MR1078484
  10. Largillier, A., 10.1016/0893-9659(94)90033-7, Appl. Math. Lett. 7 (1994), 67-71. (1994) Zbl0804.46026MR1350148DOI10.1016/0893-9659(94)90033-7
  11. Lindenstrauss, J., Preiss, D., Fréchet differentiability of Lipschitz functions (a survey), In: Recent Progress in Functional Analysis, 19-42, North-Holland Math. Stud. 189, North-Holland, Amsterdam (2001). (2001) Zbl1037.46043MR1861745
  12. Lindenstrauss, J., Preiss, D., 10.4007/annals.2003.157.257, Annals Math. 157 (2003), 257-288. (2003) Zbl1171.46313MR1954267DOI10.4007/annals.2003.157.257
  13. Preiss, D., Almost differentiability of convex functions in Banach spaces and determination of measures by their values on balls, Collection: Geometry of Banach spaces (Strobl, 1989), 237-244, London Math. Soc. Lecture Note Ser. 158 (1990). (1990) MR1110199
  14. Preiss, D., Zajíček, L., 10.1007/BF02773371, Israel J. Math. 125 (2001), 1-27. (2001) MR1853802DOI10.1007/BF02773371
  15. Veselý, L., On the multiplicity points of monotone operators on separable Banach spaces, Comment. Math. Univ. Carolin. 27 (1986), 551-570. (1986) MR0873628
  16. Veselý, L., Zajíček, L., Delta-convex mappings between Banach spaces and applications, Dissertationes Math. (Rozprawy Mat.) 289 (1989). (1989) MR1016045
  17. Zajíček, L., On the points of multivaluedness of metric projections in separable Banach spaces, Comment. Math. Univ. Carolin. 19 (1978), 513-523. (1978) MR0508958
  18. Zajíček, L., On the points of multiplicity of monotone operators, Comment. Math. Univ. Carolin. 19 (1978), 179-189. (1978) MR0493541
  19. Zajíček, L., On the differentiation of convex functions in finite and infinite dimensional spaces, Czech. Math. J. 29 (1979), 340-348. (1979) MR0536060
  20. Zajíček, L., Differentiability of the distance function and points of multi-valuedness of the metric projection in Banach space, Czech. Math. J. 33 (1983), 292-308. (1983) MR0699027
  21. Zajíček, L., On σ -porous sets in abstract spaces, Abstract Appl. Analysis 2005 (2005), 509-534. (2005) MR2201041

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.