A Gelfand-Phillips space not containing l(1) whose dual ball is not weak* sequentially compact
2001 (English)In: Glasgow Mathematical Journal, ISSN 0017-0895, Vol. 43, 125-128 p.Article in journal (Refereed) Published
A set D in a Banach space E is called limited if pointwise convergent sequences of linear functionals converge uniformly on D and E is called a GP-space (after Gelfand and Phillips) if every limited set in E is relatively compact. Banach spaces with weak* sequentially compact dual balls (W*SCDB for short) are GP-spaces and l(1)(A) is a GP-space without W*SCDB. Disproving a conjecture of Rosenthal and inspired by James tree space, Hagler and Odell constructed a class of Banach spaces ([HO]-spaces) without both W*SCDB and subspaces isomorphic to l(1). Schlumprecht has shown that there is a subclass of the [HO]-spaces which are also GP-spaces. It is not clear however if any [HO]-construction yields a GP-space-in fact it is not even clear that W*SCDB double left right arrow GP-space is false in general for the class of Banach spaces containing no subspace isomorphic to l(1). In this note the example of Hagler and Odell is modified to yield a GP-space without W*SCDB and without an isomorphic copy of l(1).
Place, publisher, year, edition, pages
2001. Vol. 43, 125-128 p.
Engineering and Technology
IdentifiersURN: urn:nbn:se:liu:diva-49294OAI: oai:DiVA.org:liu-49294DiVA: diva2:270190