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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Lineability and spaceability for the weak form of Peano's theorem and vector-valued sequence spaces


Authors: Cleon S. Barroso, Geraldo Botelho, Vinícius V. Fávaro and Daniel Pellegrino
Journal: Proc. Amer. Math. Soc. 141 (2013), 1913-1923
MSC (2010): Primary 15A03, 46B45, 34A12
Published electronically: December 28, 2012
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Abstract: Two new applications of a technique for spaceability are given in this paper. For the first time this technique is used in the investigation of the algebraic genericity property of the weak form of Peano's theorem on the existence of solutions of the ODE $ u'=f(u)$ on $ c_0$. The space of all continuous vector fields $ f$ on $ c_0$ is proved to contain a closed $ \mathfrak{c}$-dimensional subspace formed by fields $ f$ for which, except for the null field, the weak form of Peano's theorem fails to be true. The second application generalizes known results on the existence of closed $ \mathfrak{c}$-dimensional subspaces inside certain subsets of $ \ell _p(X)$-spaces, $ 0 < p < \infty $, to the existence of closed subspaces of maximal dimension inside such subsets.


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Additional Information

Cleon S. Barroso
Affiliation: Departamento de Matemática, Campus do Pici, Universidade Federal do Ceará, 60.455-760 Fortaleza, Brazil
Email: cleonbar@mat.ufc.br

Geraldo Botelho
Affiliation: Faculdade de Matemática, Universidade Federal de Uberlândia, 38.400-902, Uberlândia, Brazil
Email: botelho@ufu.br

Vinícius V. Fávaro
Affiliation: Faculdade de Matemática, Universidade Federal de Uberlândia, 38.400-902, Uberlândia, Brazil
Email: vvfavaro@gmail.com

Daniel Pellegrino
Affiliation: Departamento de Matemática, Universidade Federal da Paraíba, 58.051-900, João Pessoa, Brazil
Email: pellegrino@pq.cnpq.br, dmpellegrino@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11466-2
PII: S 0002-9939(2012)11466-2
Received by editor(s): June 2, 2011
Received by editor(s) in revised form: September 24, 2011
Published electronically: December 28, 2012
Additional Notes: The first author was supported by CNPq Grant 307210/2009-0.
The second author was supported by CNPq Grant 306981/2008-4.
The third author was supported by FAPEMIG Grant CEX-APQ-00208-09.
The fourth author was supported by CNPq Grant 301237/2009-3 and CAPES-NF
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2012 American Mathematical Society