Complutense University Library

Lineability and additivity in R(R).

Gámez Merino, José Luis and Muñoz Fernández, Gustavo Adolfo and Seoane Sepúlveda, Juan Benigno (2010) Lineability and additivity in R(R). Journal of Mathematical Analysis and Applications, 369 . pp. 265-272. ISSN 0022-247X

[img] PDF
Restricted to Repository staff only until 31 December 2020.

190kB

Official URL: http://www.sciencedirect.com/science/article/pii/S0022247X10002453

View download statistics for this eprint

==>>> Export to other formats

Abstract

We give a condition for a family of functions to be lineable by means of its additivity. The novelty presented here is that we solve the lineability problem using a technique that is not constructive, as are most approaches to this problem. We relate the notions of additivity and lineability and use this relation to give a general method to find the lineability of large families of functions. We also study more examples of pathologically behaving functions, in particular, the class of Jones functions, which is a highly pathological subclass of perfectly everywhere surjective functions. We work on the additivity, lineability, and main properties of this class.

Item Type:Article
Uncontrolled Keywords:Lineability; Additivity; Jones functions
Subjects:Sciences > Mathematics > Mathematical analysis
ID Code:15406
References:

R.M. Aron, V.I. Gurariy, J.B. Seoane-Sepulveda, Lineability and spaceability of sets of functions on R, Proc. Amer. Math. Soc. 133 (3) (2005) 795–803.

R.M. Aron, F.J. Garcia-Pacheco, D. Perez-Garcia, J.B. Seoane-Sepulveda, On dense-lineability of sets of functions on R, Topology 48 (2009) 149–156.

R.M. Aron, D. Perez-Garcia, J.B. Seoane-Sepulveda, Algebrability of the set of non-convergent Fourier series, Studia Math. 175 (1) (2006) 83–90.

F. Bayart, L. Quarta, Algebras in sets of queer functions, Israel J. Math. 158 (2007) 285–296.

L. Bernal-Gonzalez, Dense-lineability in spaces of continuous functions, Proc. Amer. Math. Soc. 136 (9) (2008) 3163–3169.

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, M. Matos, D. Pellegrino, Lineability of summing sets of homogeneous polynomials, Linear Multilinear Algebra 58 (1) (2010) 61–74.

K. Ciesielski, A.W. Miller, Cardinal invariants concerning functions whose sum is almost continuous, Real Anal. Exchange 20 (2) (1994/1995) 657–672.

K. Ciesielski, T. Natkaniec, Algebraic properties of the class of Sierpinski–Zygmund functions, Topology Appl. 79 (1) (1997) 75–99.

K. Ciesielski, I. Recław, Cardinal invariants concerning extendable and peripherally continuous functions, Real Anal. Exchange 21 (2) (1995/1996) 459– 472.

J.L. Gamez-Merino, G.A. Munoz-Fernandez, V.M. Sanchez, J.B. Seoane-Sepulveda, Sierpinski–Zygmund functions and other problems on lineability, Proc. Amer. Math. Soc., in press.

D. Garcia, B.C. Grecu, M. Maestre, J.B. Seoane-Sepulveda, Infinite dimensional Banach spaces of functions with nonlinear properties, Math. Nachr., in press.

R.G. Gibson, T. Natkaniec, Darboux like functions, Real Anal. Exchange 22 (2) (1996/1997) 492–533.

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.

S. Hencl, Isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Holder functions, Proc. Amer. Math. Soc. 128 (12) (2000) 3505–3511.

F.E. Jordan, Cardinal numbers connected with adding Darboux-like functions, Ph.D. dissertation, West Virginia University, USA, 1998.

F.B. Jones, Connected and disconnected plane sets and the functional equation f (x) + f (y) = f (x + y), Bull. Amer. Math. Soc. 48 (1942) 115–120.

K.R. Kellum, Sums and limits of almost continuous functions, Colloq. Math. 31 (1974) 125–128.

K.R. Kellum, Almost continuity and connectivity—sometimes it’s as easy to prove a stronger result, Real Anal. Exchange 8 (1) (1982/1983) 244–252.

K.R. Kellum, B.D. Garrett, Almost continuous real functions, Proc. Amer. Math. Soc. 33 (1972) 181–184.

A.B. Kharazishvili, A.B. Kirtadze, On the measurability of functions with respect to certain classes of measures, Georgian Math. J. 11 (3) (2004) 489–494.

T. Natkaniec, Almost continuity, Real Anal. Exchange 17 (2) (1991/1992) 462–520.

T. Natkaniec, New cardinal invariants in real analysis, Bull. Pol. Acad. Sci. Math. 44 (2) (1996) 251–256.

L. Rodriguez-Piazza, Every separable Banach space is isometric to a space of continuous nowhere differentiable functions, Proc. Amer. Math. Soc. 123 (12) (1995) 3649–3654.

D. Puglisi, J.B. Seoane-Sepulveda, Bounded linear non-absolutely summing operators, J. Math. Anal. Appl. 338 (1) (2008) 292–298.

J. Stallings, Fixed point theorems for connectivity maps, Fund. Math. 47 (1959) 249–263.

Deposited On:29 May 2012 09:37
Last Modified:06 Feb 2014 10:24

Repository Staff Only: item control page