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Some results and open questions on spaceability in function spaces
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Some results and open questions on spaceability in function spaces
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Journal Article
Some results and open questions on spaceability in function spaces
2014
Overview
A subset MM of a topological vector space XX is called lineable (respectively, spaceable) in XX if there exists an infinite dimensional linear space (respectively, an infinite dimensional closed linear space) Y⊂M∪{0}Y \\subset M\\cup \\{0\\}. In this article we prove that, for every infinite dimensional closed subspace XX of C[0,1]\\mathcal {C}[0,1], the set of functions in XX having infinitely many zeros in [0,1][0,1] is spaceable in XX. We discuss problems related to these concepts for certain subsets of some important classes of Banach spaces (such as C[0,1]\\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 C[0,1]\\mathcal {C}[0,1], as well as oscillating and annulling properties of subspaces of C[0,1]\\mathcal {C}[0,1].
Publisher
American Mathematical Society
