A Pythagorean inequality

Author:
Russell M. Reid

Journal:
Proc. Amer. Math. Soc. **123** (1995), 831-839

MSC:
Primary 42C30; Secondary 15A99

DOI:
https://doi.org/10.1090/S0002-9939-1995-1231041-0

MathSciNet review:
1231041

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Abstract | References | Similar Articles | Additional Information

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 ,

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 .

**[1]**R. P. Boas Jr.,*A general moment problem*, Amer. J. Math.**63**(1941), 361–370. MR**0003848**, https://doi.org/10.2307/2371530**[2]**Philip J. Davis,*Interpolation and approximation*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR**0157156****[3]**Norman Levinson,*Gap and Density Theorems*, American Mathematical Society Colloquium Publications, v. 26, American Mathematical Society, New York, 1940. MR**0003208****[4]**Laurent Schwartz,*Approximation d’un fonction quelconque par des sommes d’exponentielles imaginaires*, Ann. Fac. Sci. Univ. Toulouse (4)**6**(1943), 111–176 (French). MR**0015553****[5]**Robert M. Young,*Interpolation for entire functions of exponential type and a related trigonometric moment problem*, Proc. Amer. Math. Soc.**56**(1976), 239–242. MR**0409832**, https://doi.org/10.1090/S0002-9939-1976-0409832-3**[6]**Robert M. Young,*An introduction to nonharmonic Fourier series*, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR**591684**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1231041-0

Keywords:
Norm inequality,
Gram matrix,
nonharmonic Fourier series

Article copyright:
© Copyright 1995
American Mathematical Society