Perturbation of frames

Abstract

The question of stability plays an important role in connection with bases. That is, if {fk}∞k=1 is a basis and {gk}∞k=1 is in some sense “close” to {fk}∞k=1, does it follow that {gk}∞k=1 is also a basis? A classical result states that if {fk}∞k=1 is a basis for a Banach space X, then a sequence {gk}∞k=1 in X is also a basis if there exists a constant λ ∈]0,1[ such that (Forumala Presented). for all finite sequences of scalars {Ck}∞k=1. The result is usually attributed to Paley and Wiener [533], but it can be traced back to Neumann [524]: in fact, it is an almost immediate consequence of Theorem 2.2.3 with Ufk: = gk.