Abstract
We consider the following problem (which is a generalisation of a folklore result Proposition 1 below): given a continuous linear operator T : X -> Y, where Y is a Banach space with a (long) sub-symmetric basis, under which conditions can we find a continuous linear operator S : X -> Y such that S(B-X) contains the basis of Y. As a tool we also consider a non-separable version of Theorem 2 below: Given an infinite subset A subset of X*, under which conditions can we find a biorthogonal system in X x A of cardinality card A?