  We solve the problem of best approximation by Parseval frames to an arbitrary
frame in a subspace of an infinite dimensional Hilbert space. We explicitly
describe all the solutions and we give a criterion for uniqueness. This best
approximation problem was previously solved under an additional assumption on
the set of Parseval frames in M. Frank, V. Paulsen, T. Tiballi, Symmetric
approximation of frames and bases in Hilbert spaces, Trans. Amer. Math. Soc.
354 (2002), 777-793. Our proof relies on the geometric structure of the set of
all Parseval frames quadratically close to a given frame. In the process we
show that its connected components can be parametrized by using the notion of
index of a pair of projections, and we prove existence and uniqueness results
of best approximation by Parseval frames restricted to these connected
components.
