# Article

Full entry | PDF   (0.2 MB)
Keywords:
convex function; G\^ateaux differentiability points; Borel set; fundamental system
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 $\ell^1(\frak c)$.
References:
[1] Argyros S., Mercourakis S.: On weakly Lindelöf Banach spaces. Rocky Mountain J. Math. 23 (1993), 395-446. MR 1226181 | Zbl 0797.46009
[2] Diestel J.: Sequences and Series in Banach Spaces. Springer-Verlag (1984), New York-Berlin. MR 0737004
[3] Deville R., Godefroy G., Zizler V.: Smoothness and Renormings in Banach Spaces. Longman Scientific & Technical Essex (1993). MR 1211634 | Zbl 0782.46019
[4] Fabian M.: Gâteaux Differentiability of Convex Functions and Topology - Weak Asplund Spaces. John Wiley and Sons, Interscience (1997). MR 1461271 | Zbl 0883.46011
[5] Finet C., Godefroy G.: Biorthogonal systems and big quotient spaces. Contemporary Mathematics 85 (1989), 87-110. MR 0983383 | Zbl 0684.46016
[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
[7] Godun B.V.: On complete biorthogonal systems in a Banach space. Funkcional. Anal. i Prilozhen. 17 (1) 1-7 (1983). MR 0695091
[8] Godun B.V., Kadec M.I.: Banach spaces without complete minimal system. Functional Anal. and Appl. 14 (1980), 301-302. MR 0595733
[9] Habala P., Hájek P., Zizler V.: Introduction to Banach spaces II. Lecture Notes, Matfyzpress Prague (1996).
[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
[11] Hewitt E., Ross K.A.: Abstract Harmonic Analysis, Vol I (1963), Vol II (1970). Springer-Verlag Berlin, New York. MR 0551496
[12] Negrepontis S.: Banach spaces and Topology. Handbook of Set-Theoretic Topology (1984), North-Holland Amsterdam, New York, Oxford, Tokyo 1045-1142. MR 0776642 | Zbl 0584.46007
[13] Phelps R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics 1364, Springer-Verlag Berlin, Heidelberg (1993). MR 1238715 | Zbl 0921.46039
[14] Plichko A.N.: Banach space without a fundamental biorthogonal system. Soviet Math. Dokl. 22 (1980), 450-453. Zbl 0513.46015
[15] Rainwater J.: A class of null sets associated with convex functions on Banach spaces. Bull. Austral. Math. Soc. 42 (1990), 315-322. MR 1073653 | Zbl 0724.46017
[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
[17] Rudin W.: Fourier analysis on groups. Interscience Publishers New York (1967). MR 0152834
[18] Talagrand M.: Deux exemples de fonctions convexes. C. R. Acad. Sci. Paris, Serie A - 461 (1979), 288 461-464. MR 0527697 | Zbl 0398.46037
[19] Valdivia M.: Simultaneous resolutions of the identity operator in normed spaces. Collect. Math. (1991), 42 265-284. MR 1203185 | Zbl 0788.47024
[20] Zajíček L.: A note on partial derivatives of convex functions. Comment. Math. Univ. Carolinae 24 (1983), 89-91. MR 0703927

Partner of