DOI: https://doi.org/10.14710/jfma.v9i1.30583

ON CONDITIONS FOR WHICH A FRE’CHET SPACE NOT CONTAINING ADMITS THE GELFAND-PHILLIPS PROPERTY

*Amos Otieno Wanjara  -  Department of Mathematics and Statistics, KAIMOSI FRIENDS UNIVERSITY, Kenya, Kenya

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Abstract

If X is a Gelfand-Phillips space, then every sequence  which is equivalent to  contains an infinite subsequence  such that  is complemented in X.  is the space of all sequences  such that  as . The norm is  It is known that if every limited set of X is relatively weakly compact, then every copy of  in X contains a subspace still isomorphic to  which is complemented in X. A Banach space is said to have the Gelfand-Phillips property if every limited subset is relatively norm compact. The question whether if a Fre’chet space does not contain   implies it can admit or have the Gelfand-Phillips property is still open. The purpose of this study was to establish the conditions for which  a Fre’chet Space can admit the Gelfand-Phillips property without . We investigated the conditions on which a Fre’chet space not containing   admits the Gelfand-Phillips property. The methodology involved the use of literature study related to Fre’chet spaces and Gelfand-Phillips property. The conditions which imply that a Fre’chet space  has the Gelfand-Phillips property in spite  of it not containing   were given.  It was established that if a Fre’chet space  has a compact subspace , then it admits the Gelfand-Phillips property even if it doesn’t contain  . The results in this study have a momentous contribution in the field of Mathematical Analysis.

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Keywords: Gelfand-Phillips Property, Fre’chet Space, Limited Set, relatively norm compact

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Section: FUNDAMENTAL MATHEMATICS AND APPLICATIONS
Language : EN
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