This is an announcement for the paper “A separable Frechet space of almost universal disposition” by C. Bargetz, J. Kakol and W. Kubis.
Abstract: The Gurarii space is the unique separable Banach space $G$ which is of almost universal disposition for finite-dimensional Banach spaces, which means that for every $\epsilon>0$, for all finite-dimensional normed spaces $E\subset F$, for every isometric embedding $e: E\rightarrow G$ there exists an $\epsilon$-isometric embedding $f: F\rightarrow G$ such that $f|E=e$. We show that $G^N$ with a special sequence of semi-norms is of almost universal disposition for finite-dimensional graded Frechet spaces. The construction relies heavily on the universal operator on the Gurarii space, recently constructed by Garbulinska-Wegrzyn and the third author. This yields in particular that $G^N$ is universal in the class of all separable Frechet spaces.
The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1603.06361