Publication: Non-Lipschitz functions with bounded gradient and related problems
Loading...
Full text at PDC
Publication Date
2012-08-15
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Citations
Exportar
Abstract
Let E be a topological vector space and let us consider a property P. We say that the subset M of E formed by the vectors in E which satisfy P is μ-lineable (for certain cardinal μ, finite or infinite) if M ∪ {0} contains an infinite dimensional linear space of dimension μ. In this note we prove that there exist uncountably infinite dimensional linear spaces of functions enjoying the following properties:(1) Being continuous on [0, 1], a.e. differentiable, with a.e. bounded derivative, and not Lipschitz. (2) Differentiable in (R2)R and not enjoying the Mean Value Theorem. (3) Real valued differentiable on an open, connected, and non-convex set, having bounded gradient,non-Lipschitz, and (therefore) not verifying the Mean Value Theorem.
Description
UCM subjects
Unesco subjects
Keywords
Citation
R.M. Aron, D. García, M. Maestre, Linearity in non-linear problems, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 95 (1) (2001) 7–12.
R.M. Aron, F.J. García-Pacheco, D. Pérez-García, J.B. Seoane-Sepúlveda, On dense-lineability of sets of functions on R, Topology 48 (2009) 149–156.
R.M. Aron, V.I. Gurariy, J.B. Seoane-Sepúlveda, Lineability and spaceability of sets of functions on R, Proc. Amer. Math. Soc. 133 (3) (2005) 795–803.
R.M. Aron, D. Pérez-García, J.B. Seoane-Sepúlveda, Algebrability of the set of non-convergent Fourier series, Studia Math. 175 (1) (2006) 83–90.
R.M. Aron, J.B. Seoane-Sepúlveda, Algebrability of the set of everywhere surjective functions on C, Bull. Belg. Math. Soc. Simon Stevin 14 (1) (2007) 25–31.
P. Bandyopadhyay, G. Godefroy, Linear structures in the set of norm-attaining functionals on a Banach space, J. Convex Anal. 13 (3–4) (2006) 489–497.
C.S. Barroso, G. Botelho, V.V. Fávaro, D. Pellegrino, Lineability and spaceability for theweak form of Peano’s theorem and vectorvalued sequence spaces, Proc. Amer. Math. Soc., accepted for publication.
G. Botelho, D. Diniz, V.V. Fávaro, D. Pellegrino, Spaceability in Banach and quasi-Banach sequence spaces, Linear Algebra Appl. 434 (2011) 1255–1260.
G. Botelho, D. Diniz, D. Pellegrino, Lineability of the set of bounded linear non-absolutely summing operators, J. Math. Anal. Appl. 357 (1) (2009) 171–175.
G. Botelho, V.V. Fávaro, D. Pellegrino, J.B. Seoane-Sepúlveda, Lp [0, 1]−∪q>pLq [0, 1] is spaceable for every p > 0, Linear Algebra Appl. 436 (2012) 2963–2965.
P.H. Enflo, V.I. Gurariy, J.B. Seoane-Sepúlveda, Some results and open questions on spaceability in function spaces, Trans. Amer. Math. Soc., accepted for publication.
V.P. Fonf, V.I. Gurariy,M.I. Kadets, An infinite dimensional subspace of C[0, 1] consisting of nowhere differentiable functions, C. R. Acad. Bulgare Sci. 52 (11–12) (1999) 13–16.
J.L. Gámez-Merino, G.A. Muñoz-Fernández, D. Pellegrino, J.B. Seoane-Sepúlveda, Bounded and unbounded polynomials and multilinear forms: characterizing continuity, Linear Algebra Appl. 436 (2012) 237–242.
J.L. Gámez-Merino,G.A.Muñoz-Fernández, V.M. Sánchez, J.B. Seoane-Sepúlveda, Sierpin´ ski–Zygmund functions and other problems on lineability, Proc. Amer. Math. Soc. 138 (11) (2010) 3863–3876.
V.I. Gurariy, Subspaces and bases in spaces of continuous functions, Dokl. Akad. Nauk SSSR 167 (1966) 971–973 (in Russian).
V.I. Gurariy, Linear spaces composed of nondifferentiable functions, C. R. Acad. Bulgare Sci. 44 (1991) 13–16.
D. Kitson, R.M. Timoney, Operator ranges and spaceability, J. Math. Anal. Appl. 378 (2) (2011) 680–686.
B. Levine, D. Milman, On linear sets in space C consisting of functions of bounded variation, Comm. Inst. Sci. Math. Méc. Univ. Kharkoff [Zapiski Inst. Mat. Mech.] 16 (4) (1940) 102–105 (in Russian, with English summary).
G.A. Muñoz-Fernández, N. Palmberg, D. Puglisi, J.B. Seoane-Sepúlveda, Lineability in subsets of measure and function spaces, Linear Algebra Appl. 428 (11–12) (2008) 2805–2812.
D. Pellegrino, E.V. Teixeira, Norm optimization problem for linear operators in classical Banach spaces, Bull. Braz. Math. Soc. (N.S.) 40 (3) (2009) 417–431.
J.B. Seoane-Sepúlveda, Chaos and lineability of pathological phenomena in analysis, ProQuest LLC, Ann Arbor, MI, Ph.D. Thesis, Kent State University, 2006.
G.L. Wise, E.B. Hall, Counterexamples in Probability and Real Analysis, The Clarendon Press Oxford University Press, New York, 1993.