Monotonic norms in ordered Banach spaces

Antiproximinal Norms in Banach Spaces

We prove that every Banach space containing a complemented copy of c0 has an antiproximinal body for a suitable norm. If, in addition, the space is separable, there is a pair of antiproximinal norms. In particular, in a separable polyhedral space X, the set of all (equivalent) norms on X having an isomorphic antiproximinal norm is dense. In contrast, it is shown that there are no antiproximinal...

Common Fixed Points in Ordered Banach Spaces

In [1] Dhage, O’Regan and Agarwal introduced the class of weak isotone mappings and the class of countably condensing mappings in an ordered Banach space and they prove some common fixed point theorems for weak isotone mappings. In this paper we introduce the notion of g-weak isotone mappings which allows us to generalize some common fixed point theorems of [1]. We recall the definition of orde...

Analytic approximations of norms in separable Banach spaces

Almost Transitive and Maximal Norms in Banach Spaces

We prove that the spaces `p, 1 < p < ∞, p 6= 2, and all infinite-dimensional subspaces of their quotient spaces do not admit equivalent almosttransitive renormings. This answers a problem posed by Deville, Godefroyand Zizler in 1993. We obtain this as a consequence of a new property ofalmost transitive spaces with a Schauder basis, namely we prove that in suchspaces the unit...

Locally Uniformly Convex Norms in Banach Spaces and Their Duals

It is shown that a Banach space with locally uniformly convex dual admits an equivalent norm that is itself locally uniformly convex.