Basics in Nonlinear Geometric Analysis

Subspaces of Hilbert-generated Banach spaces and the quantification of super weak compactness

“…Notation. In general, our notation is standard and follows textbooks such as [10]. For example, if X is a Banach space, B X denotes its closed unit ball and the diameter of a bounded set A ⊂ X is denoted by diam(A).…”

$\ell_1$ spreading models and FPP in Banach spaces with monotone Schauder basis

“…In this section we set up terminology and present some basic facts. For more details we refer the reader to [2,17,43]. All Banach spaces considered here are real.…”

“…In what follows we shall use that (x n ) is weakly null to show that (s k ) is suppression 1-unconditional. The reasoning behind the arguments we put forward is classical, and was probably first used by Odell [39] (see also the proof of [17,Theorem 6.7]). Fix an integer m ≥ 2, take arbitrary scalars (a i ) m i=1 ∈ [−1, 1] m and pick any subset Λ ⊂ {1, .…”

“…There are several works concerning spreading models, as well as the behavior of semi-normalized weakly null sequences in Banach spaces (cf. [4,10,12,17] and references therein). The impact of remarkable structural properties of weakly null sequences has been a topic of great interest in Metric Fixed Point Theory.…”

Almost Surjective Epsilon-Isometry in The Reflexive Banach Spaces

“…Therefore, every reflexive normed space is Banach space. In addition, every closed subspace of reflexive Banach space is reflexive [4]. .…”