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Gelfand-Phillips Property in a Banach Space of Vector Valued Measures

✍ Scribed by G. Of Emmanuele Catania

Publisher
John Wiley and Sons
Year
1986
Tongue
English
Weight
141 KB
Volume
127
Category
Article
ISSN
0025-584X

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✦ Synopsis

In this paper (8, Z, p) will always denote a finite measure space and X a BA-NACH space. K ( p , 9) will denote the BANACH space of p-continuous X-valued measures G defined on L ? having relatively coinpact range, endowed with the semivariation norm (see

Purpose of this note is to show that if X is a BANACH space having weak* sequentially compact dual balls or a separably complemented BANACH space, then K ( p , X ) has the socalled GELFAND-PHILLIPS property (or R ( p , X ) is a GELFAWD-PHILLIPS space), i.e. any limited set in K ( p , X ) is a relatively compact set; we recall that a (bounded) set A in a BANACR space E is said to be limited if for any (x:) c E*, x,* -%, we have lim sup Is,*(x)l= 0. to* n z c A This class of BAWACH spaces has been recently investigated by BOURGAIN

In order to obtain the results above mentioned, we have to use some lemmas.

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