Complutense University Library

Lineability in subsets of measure and function spaces

Muñoz-Fernández, Gustavo Adolfo and Palmberg, M. and Puglisi, D. and Seoane Sepúlveda, Juan Benigno (2008) Lineability in subsets of measure and function spaces. Linear Algebra and its Applications, 428 (11-12). pp. 2805-2812. ISSN 0024-3795

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

135kB

Official URL: http://www.sciencedirect.com/science/article/pii/S0024379508000372

View download statistics for this eprint

==>>> Export to other formats

Abstract

We show, among other results, that if lambda denotes the Lebesgue measure on the Borel sets in [0, 1] and X is an infinite dimensional Banach space, then the set of measures whose range is neither closed nor convex is lineable in ca(lambda, X). We also show that, in certain situations, we have lineability of the set of X-valued and non-sigma-finite measures with relatively compact range. The lineability of sets of the type L-p(I)\L-q (I) is studied and some open questions are proposed. Some classical techniques together with the converse of the Lyapunov Convexity Theorem are used.


Item Type:Article
Uncontrolled Keywords:Lineability; Spaceability; Linear spaces; Measure space; Injective measure; Function spaces
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:17086
References:

A. Aizpuru, C. Pérez-Eslava, J.B. Seoane-Sepúlveda, Linear structure of sets of divergent sequences and series, Linear Algebra Appl. 418 (2–3) (2006) 595–598.

R.M. Aron, D. García, M. Maestre, Linearity in non-linear problems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 95 (2001) 7–12.

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

R. Anantharaman, J. Diestel, Sequences in the range of a vector measure, Comment. Math. Prace Mat. 30 (2) (1991) 221–235.

R.M. Aron, D. Pérez-García, J.B. Seoane-Sepúlveda, Algebrability of the set of non-convergent Fourier series, Studia Math. 175 (2006) 83–90.

J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York, 1984.

J. Diestel, J.J. Uhl Jr., Vector measures, Mathematical Surveys, American Mathematical Society, Providence, R.I. 15 (1977).

N. Dunford, J.T. Schwartz, Linear Operators. I. General Theory, Interscience Publishers Inc., New York Interscience Publishers, Ltd., London, 1958.

P. Enflo, V.I. Gurariy, On lineability and spaceability of sets in function spaces, Preprint.

V. Fonf, V.I. Gurariy, V. Kadeˇc, An infinite dimensional subspace of C[0, 1] consisting of nowhere differentiable functions, C.R. Acad. Bulgare Sci. 52 (11–12) (1999) 13–16.

D. García, B.C. Grecu, M. Maestre, J.B. Seoane-Sepúlveda, Infinite dimensional Banach spaces of functions with nonlinear properties. Preprint.

F.J. García-Pacheco, N. Palmberg, J.B. Seoane-Sepúlveda, Lineability and algebrability of pathological phenomena in analysis, J. Math. Anal. Appl. 326 (2) (2007) 929–939.

F.J. García-Pacheco, J.B. Seoane-Sepúlveda, Vector spaces of non-measurable functions, Acta Math. Sinica (English Series) 22 (6) (2006) 1805–1808.

V.I. Gurariy, L. Quarta, On lineability of sets of continuous functions, J. Math. Anal. Appl. 294 (2004) 62–72.

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

L. Janicka, N. Kalton, Vector measures of infinite variation, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (3) (1977) 239–241.

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

H.P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from Lp(ν) to Lr (ν), J. Funct. Anal. 4 (1969) 176–214.

E. Thomas, The Lebesgue–Nikodym theorem for vector valued measured, Mem. AMS 139 (1974).

W. Wnuk, The converse of Lyapunov convexity theorem. Comment, Math. Prace Mater. 21 (2) (1980) 389–390

Deposited On:13 Nov 2012 09:55
Last Modified:07 Feb 2014 09:41

Repository Staff Only: item control page