Geometry of Epimorphisms and Frames

Using a bijection between the set BH of all Bessel sequences in a (separable) Hilbert space H and the space L(`2,H) of all (bounded linear) operators from `2 to H, we endow the set F of all frames in H with a natural topology for which we determine the connected components of F . We show that each component is a homogeneous space of the group GL(`2) of invertible operators of `2. This geometrical result shows that every smooth curve in F can be lifted to a curve in GL(`2): given a smooth curve γ in F such that γ(0) = Ξ, there exists a smooth curve Γ in GL(`2) such that γ = Γ · Ξ, where the dot indicates the action of GL(`2) over F . We also present a similar study of the set of Riesz sequences.