| Title:
|
Convex functions with non-Borel set of Gâteaux differentiability points (English) |
| Author:
|
Holický, P. |
| Author:
|
Šmídek, M. |
| Author:
|
Zajíček, L. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
39 |
| Issue:
|
3 |
| Year:
|
1998 |
| Pages:
|
469-482 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show that on every nonseparable Banach space which has a fundamental system (e.g\. on every nonseparable weakly compactly generated space, in particular on every nonseparable Hilbert space) there is a convex continuous function $f$ such that the set of its G\^ateaux differentiability points is not Borel. Thereby we answer a question of J. Rainwater (1990) and extend, in the same time, a former result of M. Talagrand (1979), who gave an example of such a function $f$ on $\ell^1(\frak c)$. (English) |
| Keyword:
|
convex function |
| Keyword:
|
G\^ateaux differentiability points |
| Keyword:
|
Borel set |
| Keyword:
|
fundamental system |
| MSC:
|
46B20 |
| MSC:
|
46B26 |
| MSC:
|
46G05 |
| idZBL:
|
Zbl 0970.46026 |
| idMR:
|
MR1666778 |
| . |
| Date available:
|
2009-01-08T18:45:27Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119025 |
| . |
| Reference:
|
[1] Argyros S., Mercourakis S.: On weakly Lindelöf Banach spaces.Rocky Mountain J. Math. 23 (1993), 395-446. Zbl 0797.46009, MR 1226181 |
| Reference:
|
[2] Diestel J.: Sequences and Series in Banach Spaces.Springer-Verlag (1984), New York-Berlin. MR 0737004 |
| Reference:
|
[3] Deville R., Godefroy G., Zizler V.: Smoothness and Renormings in Banach Spaces.Longman Scientific & Technical Essex (1993). Zbl 0782.46019, MR 1211634 |
| Reference:
|
[4] Fabian M.: Gâteaux Differentiability of Convex Functions and Topology - Weak Asplund Spaces.John Wiley and Sons, Interscience (1997). Zbl 0883.46011, MR 1461271 |
| Reference:
|
[5] Finet C., Godefroy G.: Biorthogonal systems and big quotient spaces.Contemporary Mathematics 85 (1989), 87-110. Zbl 0684.46016, MR 0983383 |
| Reference:
|
[6] Godun B.V.: Biortogonal'nyje sistemy v prostranstvach ogranichennyh funkcij.Dokl. Akad. Nauk. Ukrain. SSR, Ser. A, n. 3 (1983), 7-9. MR 0698870 |
| Reference:
|
[7] Godun B.V.: On complete biorthogonal systems in a Banach space.Funkcional. Anal. i Prilozhen. 17 (1) 1-7 (1983). MR 0695091 |
| Reference:
|
[8] Godun B.V., Kadec M.I.: Banach spaces without complete minimal system.Functional Anal. and Appl. 14 (1980), 301-302. MR 0595733 |
| Reference:
|
[9] Habala P., Hájek P., Zizler V.: Introduction to Banach spaces II.Lecture Notes, Matfyzpress Prague (1996). |
| Reference:
|
[10] Haydon R.: On Banach spaces which contain $\ell^1(\tau)$ and types of measures on compact spaces.Israel J. Math 28 (1997), 313-324. MR 0511799 |
| Reference:
|
[11] Hewitt E., Ross K.A.: Abstract Harmonic Analysis, Vol I (1963), Vol II (1970).Springer-Verlag Berlin, New York. MR 0551496 |
| Reference:
|
[12] Negrepontis S.: Banach spaces and Topology.Handbook of Set-Theoretic Topology (1984), North-Holland Amsterdam, New York, Oxford, Tokyo 1045-1142. Zbl 0584.46007, MR 0776642 |
| Reference:
|
[13] Phelps R.R.: Convex Functions, Monotone Operators and Differentiability.Lecture Notes in Mathematics 1364, Springer-Verlag Berlin, Heidelberg (1993). Zbl 0921.46039, MR 1238715 |
| Reference:
|
[14] Plichko A.N.: Banach space without a fundamental biorthogonal system.Soviet Math. Dokl. 22 (1980), 450-453. Zbl 0513.46015 |
| Reference:
|
[15] Rainwater J.: A class of null sets associated with convex functions on Banach spaces.Bull. Austral. Math. Soc. 42 (1990), 315-322. Zbl 0724.46017, MR 1073653 |
| Reference:
|
[16] Rosenthal H.P.: On quasi-complemented subspaces, with an appendix on compactness of operators from $L^p(\mu)$ to $L^r(\nu)$.J. Functional Analysis 4 (1969), 176-214. MR 0250036 |
| Reference:
|
[17] Rudin W.: Fourier analysis on groups.Interscience Publishers New York (1967). MR 0152834 |
| Reference:
|
[18] Talagrand M.: Deux exemples de fonctions convexes.C. R. Acad. Sci. Paris, Serie A - 461 (1979), 288 461-464. Zbl 0398.46037, MR 0527697 |
| Reference:
|
[19] Valdivia M.: Simultaneous resolutions of the identity operator in normed spaces.Collect. Math. (1991), 42 265-284. Zbl 0788.47024, MR 1203185 |
| Reference:
|
[20] Zajíček L.: A note on partial derivatives of convex functions.Comment. Math. Univ. Carolinae 24 (1983), 89-91. MR 0703927 |
| . |
