A Pythagorean inequality
Author:
Russell M. Reid
Journal:
Proc. Amer. Math. Soc. 123 (1995), 831839
MSC:
Primary 42C30; Secondary 15A99
MathSciNet review:
1231041
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Abstract: Let be a sequence of elements of a Hilbert space, and suppose that (one or both of) the inequalities hold for every finite sequence of scalars . If an element is adjoined to , then the resulting set satisfies (one or both of) , where, denoting the norm of by r and its distance from the closed linear span of the by , and Both bounds are best possible. If is in the span of the original set, the expressions above simplify to and . If the original set is a single unit vector , so , and if is a unit vector so , then the above is , the Pythagorean Theorem. Several consequences are deduced. If are unit vectors, , and is the distance from to the span of its predecessors (so that the volume of the parallelotope spanned by the is ), the above result is used to show that .
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512310410
PII:
S 00029939(1995)12310410
Keywords:
Norm inequality,
Gram matrix,
nonharmonic Fourier series
Article copyright:
© Copyright 1995 American Mathematical Society
