  We prove that a Hilbert space frame $\fti$ contains a Riesz basis if every
subfamily $\ftj , J \subseteq I ,$ is a frame for its closed span. Secondly we
give a new characterization of Banach spaces which do not have any subspace
isomorphic to $c_0$. This result immediately leads to an improvement of a
recent theorem of Holub concerning frames consisting of a Riesz basis plus
finitely many elements.
