Lineability of non-differentiable Pettis primitives. (English) Zbl 1331.46033

The main result of the paper is the very interesting discovery that the set ND(\(X\)) of strongly measurable Pettis integrable functions on \([0,1]\) into a Banach space \(X\), with nowhere weakly differentiable primitives (w.r.t. Lebesgue measure), is indeed lineable, i.e., ND(\(X\)) contains all non-zero elements of an infinite-dimensional vector space.
A piece of history: Recall that the indefinite Bochner-integral is almost everywhere differentiable. B. J. Pettis proved in [Trans. Am. Math. Soc. 44, 277–304 (1938; Zbl 0019.41603; JFM 64.0371.02)] that this is not so for the Pettis-integral of a strongly measurable function, but asked if such a Pettis-integral is weakly almost everywhere differentiable. The question was quite dramatically solved in 1995 when S. J. Dilworth and M. Girardi [Quaest. Math. 18, No. 4, 365–380 (1995; Zbl 0856.28006)] proved that ND(\(X\)) is always non-empty.
The paper is very technical. It uses the ideas of Dilworth and Girardi [loc.cit.]together with some interesing applications of Dvoretzky’s theorem and quite a bit of infinite combinatorics. In order to help the reader, the result is first shown for \(X=\ell_2\) (Section 3) and then in general (Section 4).
The problem whether ND(\(X\)) is spaceable, i.e., contains all non-zero elements of an infinite-dimensional vector space closed in the norm of \(X\), is left open.


46G10 Vector-valued measures and integration
28B05 Vector-valued set functions, measures and integrals
15A03 Vector spaces, linear dependence, rank, lineability
Full Text: DOI arXiv


[1] Pettis, B.J.: On integration in vector spaces. Trans. Am. Math. Soc. 44, 227-304 (1938) · doi:10.1090/S0002-9947-1938-1501970-8
[2] Dilworth, S.J., Girardi, M.: Bochner vs. Pettis norm: examples and results. Banach spaces. Merida Contemp. Math. 144 American Mathematical Society Providence, RI 1993, 69-80 (1992) · Zbl 0802.46052
[3] Kadets, V.M.: Non-differentiable indefinite Pettis integral. Quaest. Math. 17(2), 137-139 (1994) · Zbl 0816.46035 · doi:10.1080/16073606.1994.9631753
[4] Munroe, M.E.: A note on weak differentiability of Pettis integrals. Bull. Am. Math. Soc. 52, 167-175 (1946) · Zbl 0061.25102 · doi:10.1090/S0002-9904-1946-08532-8
[5] Phillips, R.S.: Integration in a convex linear topological space. Trans. Am. Math. Soc. 47, 114-145 (1940) · doi:10.1090/S0002-9947-1940-0002707-3
[6] Dilworth, S.J., Girardi, M.: Nowhere weak differentiability of the Pettis integral. Quaest. Math. 18, 365-380 (1995) · Zbl 0856.28006 · doi:10.1080/16073606.1995.9631809
[7] Popa, D.: Sets which are dense in the space of all Pettis integrable functions. Quaest. Math. 23(4), 525-28 (2000) · Zbl 0974.28008 · doi:10.2989/16073600009485995
[8] Gurariy, V.I.: Subspaces and bases in spaces of continuous functions (Russian). Dokl. Akad. Nauk SSSR 167, 971-973 (1966) · Zbl 0185.20203
[9] Aron, R.M., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Lineability and spaceability of sets of functions on \[\mathbb{R}\] R. Proc. Am. Math. Soc. 133, 795-803 (2005) · Zbl 1069.26006 · doi:10.1090/S0002-9939-04-07533-1
[10] Aron, R.M., Pérez-Garcá, D., Seoane-Sepúlveda, J.B.: Algebrability of the set of non- convergent Fourier series. Studia Math. 175(1), 83-90 (2006) · Zbl 1102.42001 · doi:10.4064/sm175-1-5
[11] Gurariy, V.I., Quarta, L.: On lineability of sets of continuous functions. J. Math. Anal. Appl. 294, 62-72 (2004) · Zbl 1053.46014 · doi:10.1016/j.jmaa.2004.01.036
[12] Gurariy, V.I.: Linear spaces composed of nondifferentiable functions. C. R. Acad. Bulg. Sci. 44(5), 13-16 (1991) · Zbl 0779.26006
[13] Fonf, V., Gurariy, V.I., Kadets, M.I.: An infinite dimensional subspace of \[C[0,1]C\][0,1] consisting of nowhere differentiable functions. C. R. Acad. Bulg. Sci. 52(11-12), 13-16 (1999) · Zbl 0945.26010
[14] Rodríguez-Piazza, L.: Every separable Banach space is isometric to a space of continuous nowhere differentiable functions. Proc. Am. Math. Soc. 123(12), 3649-3654 (1995) · Zbl 0844.46007 · doi:10.2307/2161889
[15] Hencl, S.: Isometrical embeddings of separable Banach spaces into the set of nowhere approximately differentiable and nowhere H ölder functions. Proc. Am. Math. Soc. 128(12), 3505-3511 (2000) · Zbl 0956.26008 · doi:10.1090/S0002-9939-00-05595-7
[16] Gła̧b, S., Kaufmann, P.L., Pellegrini, L.: Spaceability and algebrability of sets of nowhere integrable functions. Proc. Am. Math. Soc. 141(6), 2025-2037 (2013) · Zbl 1277.26008
[17] Bernal-González, L., Ordez, M.: Cabrera, lineability criteria, with applications. J. Funct. Anal. 266(6), 3997-4025 (2014) · Zbl 1298.46024
[18] Enflo, P.H., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Some results and open questions on spaceability in function space. Trans. Am. Math. Soc. 366(2), 611-625 (2014) · Zbl 1297.46022 · doi:10.1090/S0002-9947-2013-05747-9
[19] Diestel, J., Uhl, J.J.: Vector measures. Math. Surv. 15 (1977) · Zbl 0369.46039
[20] Musial, K.: Topics in the theory of Pettis integration. Rend. Istit. Mat. Univ. Trieste 23, 177-262 (1991) · Zbl 0798.46042
[21] Singer, I.: Bases in Banach Spaces I. Springer-Verlag, Berlin (1981) · Zbl 0467.46020 · doi:10.1007/978-3-642-67844-8
[22] Dvoretzky, A.: Some results on convex bodies and Banach spaces. In: Proceedings of the International Symposium on Linear Spaces (Jerusalem, 1960), pp. 123-160. Jerusalem Academic Press, Jerusalem (1961) · Zbl 0119.31803
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.