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

 

On complete biorthogonal systems


Author: Robert M. Young
Journal: Proc. Amer. Math. Soc. 83 (1981), 537-540
MSC: Primary 42C30; Secondary 30D99
MathSciNet review: 627686
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Abstract: Fundamental to the study of bases in a separable Hilbert space $ H$ is the notion of a biorthogonal system. Two sequences $ \left\{ {{f_n}} \right\}$ and $ \left\{ {{g_n}} \right\}$ of elements from $ H$ are said to be biorthogonal if $ ({f_n},{g_m}) = {\delta _{nm}}$. A complete sequence that possesses a biorthogonal sequence is called exact. Despite the symmetry of the definition of biorthogonality, simple examples show that $ \{ {f_n}\} $ may be exact while $ \{ {g_n}\} $ fails to be exact. For sequences of complex exponentials in $ {L^2}( - \pi ,\pi )$, the situation is dramatically different--if the sequence $ \{ {e^{i{\lambda _n}t}}\} $ is exact, then its biorthogonal sequence is also exact.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0627686-9
PII: S 0002-9939(1981)0627686-9
Keywords: Biorthogonal system, exact sequence, Paley-Wiener space
Article copyright: © Copyright 1981 American Mathematical Society