Uniform Factorizations and Almost over Complete with Superprojective on Banach Spaces

Abstract

A Banach Space admits an equivalent strongly uniformly Gâteaux smooth norm if and only if it contains the dense range of a super weakly compact operator ,which is equivalent to generated by a convex super weakly compact set. We show that relatively 𝑝-compact subsets of a Banach space of 𝑝-compact operator are collectively 𝑝-compact. We study the almost over complete and almost over total sequences in Banach spaces. We also study super projective Banach spaces. We show that they cannot contain copies of ℓ1, which restricts the search for non-reflexive examples of these spaces.