The Josefson-Nissenzweig Theorem
From Alaoglu’s theorem and the F. Riesz theorem, we can conclude that for infinite-dimensional Banach spaces X the weak* topology and the norm topology in X* differ. Can they have the same convergent sequences? The answer is a resounding “no!” and it is the object of the present discussion. More precisely we will prove the following theorem independently discovered by B. Josef son and A. Nissenzweig.
KeywordsBanach Space Real Banach Space Null Sequence Unit Vector Basis Szlenk Index
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- Bourgain, J. and Diestel, J. 1984. Limited operators and strict cosingularity. Math. Nachrichten, to appear.Google Scholar
- Dineen, S. 1981. Complex Analysis in Locally Convex Spaces. North-Holland Mathematics Studies, Vol. 57. New York: North-Holland.Google Scholar
- Nissenzweig, A. 1975. ω* sequential convergence. Israel J. Math., 22, 266–272.Google Scholar