Quasi-Banach spaces of almost universal disposition

Abstract

We show that for each $p\in (0,1]$ there exists a separable $p$-Banach space ${\mathbb{G}}_p$ of almost universal disposition, that is, having the following extension property: for each $\epsilon >0$ and each isometric embedding $g:X\to Y$, where $Y$ is a finite dimensional $p$-Banach space and $X$ is a subspace of ${\mathbb{G}}_p$, there is an $\epsilon$-isometry $f:Y\to {\mathbb{G}}_p$ such that $x=f(g(x))$ for all $x\in X$. 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 injective". No separably injective $p$-Banach space was previously known for $p<1$.

Final version. Minor corrections, the proof of Corollary 6.6 expanded (24 pages)

References in corpus (2)

• Banach spaces of universal disposition
• Universal homomorphisms, universal structures, and the polymorphism clones of homogeneous structures

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• Polish spaces of Banach spaces. Complexity of isometry and isomorphism classes
• 1-complemented subspaces of Banach spaces of universal disposition