Lineability and spaceability for the weak form of Peano's theorem and vector-valued sequence spaces (Q2838949)

The authors prove two results on spaceability. First, they examine the set of vector fields \(f:\mathbb R \times c_0 \to c_0\) such that the weak form of Peano's theorem fails, namely, \(u^\prime = f(t,u)\) has no local solution. If \(\mathcal K(c_0)\) denotes this collection, they show that \(\mathcal K(c_0)\) is spaceable in the space of all continuous vector fields \(\mathbb R \times c_0 \rightarrow c_0\), \(\mathcal C(c_0)\), when this space is endowed with the topology of uniform convergence on bounded sets. In addition, several results are given which, roughly, describe the size of subspaces contained in the set difference of an \(\ell_p\)-sum of Banach spaces with the union of all \(\ell_q\)-sums, over all \(q < p\). As one corollary of their main results, they show that, for any infinite-dimensional Banach space \(X\) and any \(p \in (0, \infty)\), \(\ell_p(X) \backslash \sup_{q < p} \ell_q(X)\) is maximally spaceable. An interesting feature of the arguments is a variation on the ``mother vector'' technique of finding one vector in the set in question and perturbing it (a lot) to get large spaces contained in the set.