The finite-dimensional decomposition property in non-Archimedean Banach spaces.

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property (OFDDP) if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces. This property has an influence in the non-Archimedean Grothendieck's approximation theory, where an open problem is the following: Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E. Does D have the OFDDP? In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP. Next we prove that, however, for certain classes of Banach spaces of countable type, the OFDDP is preserved by taking finite-codimensional subspaces. [ABSTRACT FROM AUTHOR]