Infinite dimensional spaces consisting of sequences that do not converge to zero

TL;DR

This paper explores the structure of infinite dimensional Banach spaces of sequences that do not converge to zero, providing new insights and results for spaces lacking classical properties and for various classes of maps.

Contribution

It introduces conditions for the existence of such spaces within Banach spaces and lattices, extending and improving known results in the field.

Findings

Existence of infinite dimensional Banach spaces of sequences not converging to zero.

Applications to spaces and maps lacking classical properties.

Enhanced understanding of sequence behavior under nonlinear and linear maps.

Abstract

Given a map $f \colon E \longrightarrow F$ between Banach spaces (or Banach lattices), a set $A$ of $E$-valued bounded sequences, ${\bf x} \in A$ and a vector topology $\tau$ on $F$, we investigate the existence of an infinite dimensional Banach space (or Banach lattice) containing a subsequence of ${\bf x}$ and consisting, up to the origin, of sequences $(x_j)_{j=1}^\infty$ belonging to $A$ such that $(f(x_j))_{j=1}^\infty$ does not converge to zero with respect to $\tau$. The applications we provide encompass the improvement of known results, as well as new results, concerning Banach spaces/Banach lattices not satisfying classical properties and linear/nonlinear maps not belonging to well studied classes.