A norm |⋅| on a Banach space X is locally uniformly rotund (LUR) if lim |xn — x| = 0 for every xn, x ∈ X for which lim2|x|2 + 2 |xn|2-|xn+xn|2 = 0. It is shown that a Banach space X admits an equivalent LUR norm provided there is a bounded linear operator T of X into c0(Γ) such that T* c*0(Γ) is norm dense in X*. This is the case e.g. if X* is weakly compactly generated (WCG).

Let d(n;l,k) denote the number of divisors of the positive integer n which are congruent to I modulo k. The objective of the present paper is to prove that (for some exponent θ<⅓)

holds uniformly in l, k and x satisfying 1≤l≤k≤x. This improves a recent result due to R. A. Smith and M. V. Subbarao [3].

Compactifications are constructed for convergence ordered spaces and topological ordered spaces with extension properties that resemble those of the Stone-Čech compactification.

In this paper, we prove that a real valued bounded function, defined on a metric space and uniformly continuous is the uniform limit of a sequence of Lipschitzian bounded functions.

As a consequence, a new criterion for the weak convergence of probabilities is given.

Let B be a Douglas algebra which admits best approximation. It will be shown that the following are equivalent: (1) The unit ball of (L∞/B) has no extreme points; (2) For any Blaschke product b with , there exists h ∈ B such that = 1 and h|E≢0, where E is the essential set of B.

It will also be proven that if B⊇H∞+C and its essential set E contains a closed Gδ set, then the unit ball of (L∞/B) has no extreme points. Many known results concerning this subject will follow from these results.