Rod Nillsen
Mathematics research
Extreme points and Banach limits (1976)
A Banach limit is a bounded linear functional on the Banach space of all bounded real-valued sequences, in the uniform norm. To describe them, if x is a bounded sequence, written as x=(x(n)), let T(x) be the sequence y, where y(n)=x(n+1), for all n=1,2,3,......Then a Banach limit is a bounded linear functional L with the properties: L is positive , L(1)=1 and L(T(x))=L(x), for all bounded sequences x. If L is a Banach limit, and x is a convergent sequence, then L(x) equals the ordinary limit of the sequence.
In the weak-star topology, the Banach limits form a compact convex set, and so must have a non-empty set of extreme points, by the Krein-Milman Theorem. It is proved in a paper published in Proc. Amer. Math. Soc., volume 55 (1976) pp.347-352, that the set of extreme points of the Banach limits is non-compact. Related work was published independently at about the same time by M. Talagrand.
October 2005