20 research outputs found
Approximation of norms on Banach spaces
Relatively recently it was proved that if is an arbitrary set, then
any equivalent norm on can be approximated uniformly on bounded
sets by polyhedral norms and smooth norms, with arbitrary precision.
We extend this result to more classes of spaces having uncountable symmetric
bases, such as preduals of the `discrete' Lorentz spaces , and
certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an
arbitrary ordinal number , there exists a scattered compact space
having Cantor-Bendixson height at least , such that every equivalent
norm on can be approximated as above
Strongly extreme points and approximation properties
We show that if is a strongly extreme point of a bounded closed convex
subset of a Banach space and the identity has a geometrically and topologically
good enough local approximation at , then is already a denting point. It
turns out that such an approximation of the identity exists at any strongly
extreme point of the unit ball of a Banach space with the unconditional compact
approximation property. We also prove that every Banach space with a Schauder
basis can be equivalently renormed to satisfy the sufficient conditions
mentioned. In contrast to the above results we also construct a non-symmetric
norm on for which all points on the unit sphere are strongly extreme, but
none of these points are denting.Comment: 14 page