A separable Fréchet space of almost universal disposition

The Gurari\uı 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 Fréchet spaces. The construction relies heavily on the universal operator on the Gurari\uı space, recently constructed by Garbulińska-Wegrzyn and the third author. This yields in particular that \(\mathbb{G}^{\mathbb{N}}\) is universal in the class of all separable Fréchet spaces.