International Journal of Nonlinear Analysis and Applications
On the differentiability of norms in Banach spaces
Document Type : Research Paper
Author
Faculty of Basic Sciences, Santiago de Cali University, Valle del Cauca, Cali, Colombia
Abstract
The purpose of this paper is to show some particularities that the differentiability sets generated from the norms have in the Banach spaces. In this sense, it will be shown that the Gaussian measure of the Fr'echet differentiability set of the norm of the space $\ell^{\infty}(\mathbb{R})$ of real bounded sequences is zero and that in the case of the space $BV[a,b]$ of bounded variation functions its norm is not Fr'echet derivable in any element of this space.
Keywords
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[8] R.R. Phelps, Gaussian null sets and differentiability of Lipschitz map on Banach spaces, Pacific Juornal of Mathematics 77 (1978), no. 2, 523–531.
[9] R.R. Phelps, Convex functions, monotone operators and differentiability, Springer, 1989.
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p norms, Advances in Mathematics 262 (2014), 436–475.
[11] H. Rademacher, Uber partielle und totale differenzierbarkeit von funktionen mehrerer variabeln und ¨uber die ¨
transformation der doppelintegrale, Math. Ann. 79 (1919), 340–359.
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[13] Z. Semadeni, Banach spaces of continuous functions, PWN, Warszawa, 1971.
[14] W. Schirotzek, Nonsmooth analysis, Springer, 2007.
[15] J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential
Equations 246 (2009), no. 7, 2788–2812.
[16] C. Stegall, The duality between Asplund spaces and spaces with the Radon-Nykod´ym property, Israel J. Math. 29
(1978), 408–412.
[17] N.N. Vakhania, Probability distributions on linear spaces, Elsevier North Holland, 1981.
Limited, 1993.
[2] M. Dore and O. Maleva, A universal differentiability set in Banach spaces with separable dual, J. Funct. Anal.
261 (2011), no. 6, 1674–1710.
[3] M. Fabian, Gateaux differentiability of convex functions and topology: weak Asplund spaces, John Wiley and Sons,
1997.
[4] M. Gieraltowska and F.S. Van Vleck, Fr´echet vs. Gˆateaux differentiability of Lipschitzian functions, Amer. Math.
Soc. 114 (1992), no. 4, 905–907.
[5] B. Goldys and F. Gozzi, Second order parabolic Hamilton- Jacobi-Bellman equations in Hilbert spaces and stochastic control: l
2
µ approach, Stochastic Process. 12 (2006), 1932–1963.
[6] J. Lindenstrauss, D. Preiss and J. Tisˇer, Fr´echet differentiability of Lipschitz functions via a variational principle,
J. Eur. Math. Soc. 12 (2010), no. 2, 385–412.
[7] I. Namioka and R.R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975), no. 4, 735–750.
[8] R.R. Phelps, Gaussian null sets and differentiability of Lipschitz map on Banach spaces, Pacific Juornal of Mathematics 77 (1978), no. 2, 523–531.
[9] R.R. Phelps, Convex functions, monotone operators and differentiability, Springer, 1989.
[10] D. Potapov and F. Sukoshev, Fr´echet differentiability of S
p norms, Advances in Mathematics 262 (2014), 436–475.
[11] H. Rademacher, Uber partielle und totale differenzierbarkeit von funktionen mehrerer variabeln und ¨uber die ¨
transformation der doppelintegrale, Math. Ann. 79 (1919), 340–359.
[12] G. Restrepo, Differentiable norms in Banach spaces, Bull. Am. Math. Soc. 70 (1964), no. 3, 413–414.
[13] Z. Semadeni, Banach spaces of continuous functions, PWN, Warszawa, 1971.
[14] W. Schirotzek, Nonsmooth analysis, Springer, 2007.
[15] J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential
Equations 246 (2009), no. 7, 2788–2812.
[16] C. Stegall, The duality between Asplund spaces and spaces with the Radon-Nykod´ym property, Israel J. Math. 29
(1978), 408–412.
[17] N.N. Vakhania, Probability distributions on linear spaces, Elsevier North Holland, 1981.
)
Volume 13, Issue 2
July 2022Pages 2015-2023
- Receive Date: 14 October 2020
- Revise Date: 16 December 2020
- Accept Date: 26 December 2020