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A Gelfand‐Phillips Property with Respect to the Weak Topology
Author(s) -
Leung Denny H.
Publication year - 1990
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19901490114
Subject(s) - mathematics , banach space , separable space , tensor product , type (biology) , compact open topology , compact space , weak topology (polar topology) , linear subspace , pure mathematics , space (punctuation) , bounded operator , product topology , discrete mathematics , topological space , interpolation space , mathematical analysis , general topology , functional analysis , extension topology , computer science , ecology , biochemistry , chemistry , gene , biology , operating system
We consider a Gelfand‐Phillips type property for the weak topology. The main results that we obtain are (1) for certain Banach spaces, E ˜ ϵ F inherits this property from E and F , and (2) the spaces L p (μ, E ) have this property when E does. A subset A of a Banach space E is a limited set if every (bounded linear) operator T:E → c 0 maps A onto a relatively compact subset of c 0 . The Banach space E has the Gelfand‐Phillips property if every limited set is relatively compact. In this note, we study the analogous notions set in the weak topology. Thus we say that A ⊂ E is a Grothendieck set if every T: E → c 0 maps A onto a relatively weakly compact set; and E is said to have the weak type GP property if every Grothendieck set in E is relatively weakly compact. In the papers [3, 4 and 6], it is shown among other results that the ϵ‐tensor product E and the spaces L p (μ, E ) inherit the Gelfand‐Phillips property from E and F. In this paper, we study the same questions for the weak type GP property. It is easily verified that continuous linear images of Grothendieck sets are Grothendieck and that the weak type GP property is inherited by subspaces. Among the spaces with the weak type GP property one easily finds the separable spaces, and more generally, spaces with a weak* sequentially compact dual ball. Also, C(K) spaces where K is (DCSC) are weak type GP (see [3] and the discussion before Corollary 4 below). A Grothendieck space (a Banach space whose unit ball is a Grothendieck set) has the weak type GP if and only if it is reflexive.