On Universal Schauder Bases in Non-Archimedean Frechet Spaces - Śliwa, Wiesław (Canadian Mathematical Society, 2004)
We prove that there exists a non-archimedean Frechet space U with a basis $(u_n)$ such that any asis $(x_n)$ in a non-archimedean Frechet space X is equivalent to a subbasis $(u_k_n)$ of $(u_n)$. Then any non-archimedean Frechet space with a basis is isomorphic to a complemented subspace of U. Next we prove that there is no nuclear non-archimedean Frechet space H with a basis $(h_n)$ such that any basis $(y_n)$ in a nuclear non-archimedean Frechet space Y is equivalent to a subbasis $(h_k_n)$ of $(h_n)$.
Keywords: Universal bases; Complemented subspaces with bases