Banach Limits Revisited

Generalizations of Banach-hausdorff Limits

In a recent paper [l],1 W. F. Eberlein introduced the notion of Banach-Hausdorff limits. We employ throughout m to denote the space of all real bounded sequences [2, pp. 11 and 34]. The BanachHausdorff limits are real-valued functionals L(x), defined over m, which are Banach limits [2, p. 34], i.e., which satisfy the four conditions (i) L(ax+by)=aL(x)+bL(y) (a, b real), (ii)L(l) = l, (iii) L(x)...

Nets of Extreme Banach Limits

Let N be the set of natural numbers and let a: N -* N be an injection having no periodic points. Let M0 be the set of o-invariant means on lx. When / G lx let da(f) = sup X(/), where the supremum is taken over all X G M„. It is shown that when/ e lx, there is a sequence (\)TM=2 of extreme points of M„ which has no extreme weak* limit points and such that \s(f) = dQ(f) for s = 2, 3, .... As a co...

Term Limits Revisited

Banach-Steinhaus theory revisited: Lineability and spaceability

In this paper we study the divergence behavior of linear approximation processes in general Banach spaces. We are interested in the structure of the set of divergence creating functions. The Banach–Steinhaus theory gives some information about this set, however, it cannot be used to answer the question whether this set contains subspaces with linear structure. We give necessary and sufficient c...

Hahn-Banach extension theorems for multifunctions revisited

Several generalizations of the Hahn–Banach extension theorem to K-convex multifunctions were stated recently in the literature. In this note we provide an easy direct proof for the multifunction version of the Hahn–Banach–Kantorovich theorem and show that in a quite general situation it can be obtained from existing results. Then we derive the Yang extension theorem using a similar proof as wel...