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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A representation theorem for Schauder bases
in Hilbert space


Authors: Stephane Jaffard and Robert M. Young
Journal: Proc. Amer. Math. Soc. 126 (1998), 553-560
MSC (1991): Primary 46B15; Secondary 47A55
MathSciNet review: 1425127
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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

\begin{displaymath}f=\sum c_nf_n.\end{displaymath}

If, in addition, there exist positive constants $A$ and $B$ 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 $H$ 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.


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Additional Information

Stephane Jaffard
Affiliation: Centre de Mathématiques et Leurs Applications, 61 Avenue du Président Wilson, 94235 Cachan cedex, France

Robert M. Young
Affiliation: Department of Mathematics, Oberlin College, Oberlin, Ohio 44074

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04168-9
PII: S 0002-9939(98)04168-9
Keywords: Schauder basis, Riesz basis.
Received by editor(s): April 17, 1996
Received by editor(s) in revised form: August 22, 1996
Communicated by: Dale Alspach
Article copyright: © Copyright 1998 American Mathematical Society