A Gelfand-Phillips property with respect to the weak topology. (English) Zbl 0765.46007

Leung, Denny H.

Math. Nachr. 149, 177-181 (1990).

A subset \(A\) of a Banach space \(E\) is called a Grothendieck set if every \(T: E\to c_ 0\) maps \(A\) onto a relatively weakly compact set. The space \(E\) is said to have the weak type Gelfand-Phillips property if every Grothendieck set in \(E\) is relatively weakly compact. The paper investigates the questions whether, when the spaces \(E\), \(F\) have this property, the \(\varepsilon\)-tensor product of \(E\) and \(F\), and \(L^ p(\mu,E)\) also have the property.

Reviewer: S.Swaminathan (Halifax)

MSC:

46B22	Radon-Nikodým, Kreĭn-Milman and related properties
46B28	Spaces of operators; tensor products; approximation properties
46M05	Tensor products in functional analysis

Keywords:

Grothendieck set; weak type Gelfand-Phillips property; relatively weakly compact; \(\varepsilon\)-tensor product

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