Local theory of frames and Schauder bases for Hilbert space

We develope a local theory for frames on finite-dimensional Hilbert spaces. We show that for every frame (f(i))(i=1)(m) for an n-dimensional Hilbert space, and for every epsilon > 0, there is a subset I subset of {1,2....,m} with \I\ greater than or equal to (1 - epsilon)n so that (f(i))(i is an element of I) is a Riesz basis for its span with Riesz basis constant a function of epsilon, the frame bounds, and (\\f(i)\\)(i=1)(m), but independent of m and n. we also construct an example of a normalized frame for a Hilbert space H which contains a subset which forms a Schauder basis for H, but contains no subset which is a Riesz basis for H. We give examples to show that all of our results are best possible, and that all parameters are necessary.