Hilbert Space Frames Containing a Riesz Basis and Banach Spaces Which Have No Subspace Isomorphic to c0

We prove that a Hilbert space frame fi4igI contains a Riesz basis if every subfamily fi4igJ, J: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 c0. This result immediately leads to an improvement of a recent theorem of Holub concerning frames consisting of a Riesz basis plus finitely many elements