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Some results and open questions on spaceability in function spaces


Authors: Per H. Enflo, Vladimir I. Gurariy and Juan B. Seoane-Sepúlveda
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
Published electronically: July 26, 2013
MathSciNet review: 3130310
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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]$.


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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
Email: jseoane@mat.ucm.es

DOI: https://doi.org/10.1090/S0002-9947-2013-05747-9
Keywords: Lineability, spaceability, subspaces of continuous functions, zeros of functions, M\"untz spaces
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.
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).
Article copyright: © Copyright 2013 American Mathematical Society

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