A separable Fréchet space of almost universal disposition

The GurariÄ­ space is the unique separable Banach space G which is of almost universal disposition for finite-dimensional Banach spaces, which means that for every ε>0, for all finite-dimensional normed spaces E⊆F, for every isometric embedding e:E→G there exists an ε-isometric embedding f:F→G such that f↾E=e. We show that GN 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 GurariÄ­ space, recently constructed by Garbulińska-Węgrzyn and the third author. In addition, we consider a non-graded sequence of semi-norms on GN with which the space GN is of almost universal disposition for finite-dimensional Fréchet spaces with a fixed sequence of semi-norms. In both cases, this yields in particular that GN is universal in the class of all separable Fréchet spaces.