A separable Fr\'echet space of almost universal disposition

TL;DR

This paper extends the concept of the Gurarii space to a separable Fre9chet space, demonstrating its almost universal disposition and universality among separable Fre9chet spaces using a novel construction.

Contribution

It constructs a separable Fre9chet space of almost universal disposition, generalizing the Gurarii space to a broader class of spaces.

Findings

Ge9G^{\u00a0}Ne9 is of almost universal disposition for finite-dimensional graded Fre9chet spaces.

Ge9G^{\u00a0}Ne9 is universal among all separable Fre9chet spaces.

Abstract

The Gurari\u{\i} space is the unique separable Banach space $\mathbb{G}$ which is of almost universal disposition for finite-dimensional Banach spaces, which means that for every $\varepsilon>0$, for all finite-dimensional normed spaces $E \subseteq F$, for every isometric embedding ${e}\colon{E}\to{\mathbb{G}}$ there exists an $\varepsilon$-isometric embedding ${f}\colon{F}\to{\mathbb{G}}$ such that $f \restriction E = e$. We show that $\mathbb{G}^{\mathbb{N}}$ with a special sequence of semi-norms is of almost universal disposition for finite-dimensional graded Fr\'echet spaces. The construction relies heavily on the universal operator on the Gurari\u{\i} space, recently constructed by Garbuli\'nska-Wegrzyn and the third author. This yields in particular that $\mathbb{G}^{\mathbb{N}}$ is universal in the class of all separable Fr\'echet spaces.