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.
Categories: math.FA