A characterization of F-spaces containing an isomorph of _0 - Fotiy, O.; Popov, M.; Ukrainets, O. (Carpathian Mathematical Publications, 2025)
We prove that an F-space X contains an isomorph of _0 if and only if there exists a continuous at zero function T L_0 → X with T0 = 0 possessing the following two properties. (1) (ε > 0)(δ > 0)(x L_0) (a > 0) (b > 0) ( \|a Tx \| ε \|T(bx) \| δ ), which is a weak version of the positive homogeneity. (2) For every x,y,z L_0 with Tx eq 0, if x is a disjoint sum of y and z, then either Ty eq 0 or Tz eq 0. We also provide examples showing that all assumptions on T are essential.