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A representation theorem for Schauder bases in Hilbert space

Authors:	Stephane Jaffard Robert M Young

Institution:	Centre de Mathématiques et Leurs Applications, 61 Avenue du Président Wilson, 94235 Cachan cedex, France Robert M. Young ; Department of Mathematics, Oberlin College, Oberlin, Ohio 44074

Abstract:	A sequence of vectors $\{f_1,f_2,f_3,\dotsc\}$ in a separable Hilbert space is said to be a Schauder basis for if every element $f\in H$ has a unique norm-convergent expansion $\begin{displaymath}f=\sum c_nf_n.\end{displaymath}$ If, in addition, there exist positive constants and such that $\begin{displaymath}A\sum\|c_n\|^2\le\left\\|\sum c_nf_n\right\\|^2\le B\sum\|c_n\|^2,\end{displaymath}$ 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 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.

Keywords:	Schauder basis Riesz basis

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