A representation theorem for Schauder bases in Hilbert space

Abstract:

A sequence of vectors $\{f_1,f_2,f_3,\dotsc \}$ in a separable Hilbert space $H$ is said to be a Schauder basis for $H$ if every element $f\in H$ has a unique norm-convergent expansion \[ f=\sum c_nf_n.\] If, in addition, there exist positive constants $A$ and $B$ such that \[ A\sum |c_n|^2\le \left \|\sum c_nf_n\right \|^2\le B\sum |c_n|^2,\] then we call $\{f_1,f_2,f_3,\dotsc \}$ a Riesz basis. In the first half of this paper, we show that every Schauder basis for $H$ can be obtained from an orthonormal basis by means of a (possibly unbounded) one-to-one positive self adjoint operator. In the second half, we use this result to extend and clarify a remarkable theorem due to Duffin and Eachus characterizing the class of Riesz bases in Hilbert space.