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arXiv · 2505.03041

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

Mikaela Aires·Geraldo Botelho
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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.

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Mikaela Aires, Geraldo Botelho. 2025-05-05. Infinite dimensional spaces consisting of sequences that do not converge to zero. https://arxiv.org/abs/2505.03041

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