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On Bessel and Grüss inequalities for orthonormal families in inner product spaces. (English) Zbl 1069.26019

Let H be an inner product space and (e i ) iI be a finite family of orthonormal vectors in H· It is proved that for all xH and all families (λ i ) I and (μ i ) I of scalars such that

Rex- iI λ i e i ,x- iI μ i e i ,0

we have

x 2 - iI x,e i 2 1 4 iI λ i -μ i 2 - iI λ i +μ i 2-x,e i 2 ·

Moreover, the constant 1/4 is sharp. This is used to prove a refinement of the Grüss inequality.

26D15Inequalities for sums, series and integrals of real functions
46C05Hilbert and pre-Hilbert spaces: geometry and topology