Lineability and spaceability of sets of functions on $\mathbb {R}$
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- by Richard Aron, V. I. Gurariy and J. B. Seoane PDF
- Proc. Amer. Math. Soc. 133 (2005), 795-803 Request permission
Abstract:
We show that there is an infinite-dimensional vector space of differentiable functions on $\mathbb {R},$ every non-zero element of which is nowhere monotone. We also show that there is a vector space of dimension $2^c$ of functions $\mathbb {R} \to \mathbb {R},$ every non-zero element of which is everywhere surjective.References
- Richard Aron, Raquel Gonzalo, and Andriy Zagorodnyuk, Zeros of real polynomials, Linear and Multilinear Algebra 48 (2000), no. 2, 107–115. MR 1813438, DOI 10.1080/03081080008818662
- P. Enflo, V. I Gurariy, On lineability and spaceability of sets in function spaces, to appear.
- V. P. Fonf, V. I. Gurariy, and M. I. Kadets, An infinite dimensional subspace of $C[0,1]$ consisting of nowhere differentiable functions, C. R. Acad. Bulgare Sci. 52 (1999), no. 11-12, 13–16. MR 1738120
- Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in analysis, The Mathesis Series, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. MR 0169961
- V. I. Gurariĭ, Subspaces and bases in spaces of continuous functions, Dokl. Akad. Nauk SSSR 167 (1966), 971–973 (Russian). MR 0199674
- V. I. Gurariĭ, Subspaces of differentiable functions in the space of continuous functions, Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 4 (1967), 116–121 (Russian). MR 0220048
- V. I. Gurariĭ, Linear spaces composed of everywhere nondifferentiable functions, C. R. Acad. Bulgare Sci. 44 (1991), no. 5, 13–16 (Russian). MR 1127022
- V. I. Gurariy, W. Lusky, Geometry of Müntz spaces, Lecture Notes in Mathematics, Springer-Verlag, to appear.
- Stanislav Hencl, Isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3505–3511. MR 1707147, DOI 10.1090/S0002-9939-00-05595-7
- Y. Katznelson and Karl Stromberg, Everywhere differentiable, nowhere monotone, functions, Amer. Math. Monthly 81 (1974), 349–354. MR 335701, DOI 10.2307/2318996
- H. Lebesgue, Leçons sur l’intégration, Gauthier-Villars (1904).
- L. Rodríguez-Piazza, Every separable Banach space is isometric to a space of continuous nowhere differentiable functions, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3649–3654. MR 1328375, DOI 10.1090/S0002-9939-1995-1328375-8
Additional Information
- Richard Aron
- Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44242
- MR Author ID: 27325
- Email: aron@math.kent.edu
- V. I. Gurariy
- Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44242
- Email: gurariy@math.kent.edu
- J. B. Seoane
- Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44242
- MR Author ID: 680972
- Email: jseoane@math.kent.edu
- Received by editor(s): March 26, 2003
- Received by editor(s) in revised form: October 28, 2003
- Published electronically: August 24, 2004
- Additional Notes: The author thanks Departamento de Matemáticas of the Universidad de Cádiz (Spain), especially Antonio Aizpuru, Fernando León, Javier Pérez, and the rest of the members of the group FQM-257.
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 795-803
- MSC (2000): Primary 26A27, 46E10, 46E15
- DOI: https://doi.org/10.1090/S0002-9939-04-07533-1
- MathSciNet review: 2113929