Some results and open questions on spaceability in function spaces
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- by Per H. Enflo, Vladimir I. Gurariy and Juan B. Seoane-Sepúlveda PDF
- Trans. Amer. Math. Soc. 366 (2014), 611-625 Request permission
Abstract:
A subset $M$ of a topological vector space $X$ is called lineable (respectively, spaceable) in $X$ if there exists an infinite dimensional linear space (respectively, an infinite dimensional closed linear space) $Y \subset M\cup \{0\}$. In this article we prove that, for every infinite dimensional closed subspace $X$ of $\mathcal {C}[0,1]$, the set of functions in $X$ having infinitely many zeros in $[0,1]$ is spaceable in $X$. We discuss problems related to these concepts for certain subsets of some important classes of Banach spaces (such as $\mathcal {C}[0,1]$ or Müntz spaces). We also propose several open questions in the field and study the properties of a new concept that we call the oscillating spectrum of subspaces of $\mathcal {C}[0,1]$, as well as oscillating and annulling properties of subspaces of $\mathcal {C}[0,1]$.References
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Additional Information
- Per H. Enflo
- Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
- Email: enflo@math.kent.edu
- Vladimir I. Gurariy
- Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
- Juan B. Seoane-Sepúlveda
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid, 28040, Spain
- MR Author ID: 680972
- Email: jseoane@mat.ucm.es
- Received by editor(s): July 15, 2011
- Received by editor(s) in revised form: October 20, 2011
- Published electronically: July 26, 2013
- Additional Notes: The second author was supported by the Spanish Ministry of Science and Innovation, grant MTM2009-07848.
- © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 366 (2014), 611-625
- MSC (2010): Primary 15A03, 26A15, 46E15
- DOI: https://doi.org/10.1090/S0002-9947-2013-05747-9
- MathSciNet review: 3130310
Dedicated: This work was completed after the passing of the second author. The first and third authors wish to dedicate this article to the loving memory of their friend and colleague, Vladimir I. Gurariy (1935-2005).