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

TL;DR

The paper addresses when sets of Banach-valued sequences that do not converge to zero under a weaker topology $\tau$ can be contained in infinite-dimensional subspaces, introducing the concept of almost pointwise spaceability between spaceability and pointwise spaceability. It proves a general theorem: for a map of homogeneous type $f:E\to F$, a subsequence-invariant and $\ell_\infty$-complete $A\subset \ell_\infty(E)$, and a vector topology $\tau$ weaker than the norm, the set $\mathcal C=\{(x_j)\in A:\ (f(x_j))_{j}\not\to_\tau 0\}$ is empty or almost pointwise spaceable in $\ell_\infty(E)$ (and similarly in $c_0^w(E)$ under appropriate intersections). This framework yields numerous concrete applications: for Banach spaces, it produces almost pointwise spaceability results for sets associated with non-completely continuous operators, non-weakly sequentially continuous polynomials, Grothendieck-type failures, and weak$^*$ phenomena; for Banach lattices, it yields almost positive pointwise latticeability results for disjoint or order-bounded sequences and identifies operator classes failing standard properties. Several results improve or extend existing lineability/spaceability outcomes, providing constructive infinite-dimensional subspaces or sublattices embedded in sets defined by nonconvergence under $\tau$.

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 $τ$ 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 $τ$. 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.

Theorems & Definitions (40)

• Definition 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Theorem 2.5
• proof
• Corollary 2.6
• Example 2.7
• Proposition 2.8
• proof