Quasi-Banach spaces of almost universal disposition

摘要

We show that for each p ∈ (0,1] there exists a separable p-Banach space G_p of almost universal disposition, that is, having the following extension property: for each ε > 0 and each isometric embedding g: X → Y, where Y is a finite-dimensional p-Banach space and X is a subspace of G_p, there is an ε-isometry f: Y → G_p such that x = f (g(x)) for all x ∈ X. Such a space is unique, up to isometries, does contain an isometric copy of each separable p-Banach space and has the remarkable property of being "locally injective" amongst p-Banach spaces. We also present a nonseparable generalization which is of universal disposition for separable spaces and "separably in-jective". No separably injective p-Banach space was previously known for p < 1.