A note on biorthogonal systems

Abstract

We consider the following problem (which is a generalisation of a folklore result Proposition 1 below): given a continuous linear operator $$T:X\rightarrow Y$$ , where Y is a Banach space with a (long) sub-symmetric basis, under which conditions can we find a continuous linear operator $$S:X\rightarrow Y$$ 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 $$A\subset X^*$$ , under which conditions can we find a biorthogonal system in $$X\times A$$ of cardinality $${{\,\mathrm{card}\,}}A$$ ?