A well-known theorem by H. Corson states that if a Banach space admits a locally finite covering by bounded closed convex subsets, then it contains no infinite-dimensional reflexive subspace. We strengthen this result proving that if an infinite-dimensional Banach space admits a locally finite covering by bounded -closed subsets, then it is -saturated, thus answering a question posed by V. Klee concerning locally finite coverings of spaces. Moreover, we provide information about massiveness of the set of singular points in (PC) spaces.
Covering a Banach space / V.P. Fonf, C. Zanco. - In: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9939. - 134:9(2006), pp. 2607-2611. [10.1090/S0002-9939-06-08254-2]
Covering a Banach space
C. ZancoUltimo
2006
Abstract
A well-known theorem by H. Corson states that if a Banach space admits a locally finite covering by bounded closed convex subsets, then it contains no infinite-dimensional reflexive subspace. We strengthen this result proving that if an infinite-dimensional Banach space admits a locally finite covering by bounded -closed subsets, then it is -saturated, thus answering a question posed by V. Klee concerning locally finite coverings of spaces. Moreover, we provide information about massiveness of the set of singular points in (PC) spaces.
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