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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.

MSC:

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

[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
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