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Zbl 1069.26019
Dragomir, S.S.
On Bessel and Grüss inequalities for orthonormal families in inner product spaces.
(English)
[J] Bull. Aust. Math. Soc. 69, No. 2, 327-340 (2004). ISSN 0004-9727

Let $H$ be an inner product space and $(e_{i})_{i\in I}$ be a finite family of orthonormal vectors in $H.$ It is proved that for all $x\in H$ and all families $(\lambda_{i})_{\in I}$ and $(\mu_{i})_{\in I}$ of scalars such that $$\text {Re}\left\langle x-\sum_{i\in I}\lambda_{i}e_{i},x-\sum_{i\in I} \mu_{i}e_{i},\right\rangle\leq0$$ we have $$\left\Vert x\right\Vert ^{2}-\sum_{i\in I}\left\vert \langle x,e_{i} \rangle\right\vert ^{2}\leq\frac{1}{4}\sum_{i\in I}\left\vert \lambda_{i} -\mu_{i}\right\vert ^{2}-\sum_{i\in I}\left\vert \frac{\lambda_{i}+\mu_{i}} {2}-\langle x,e_{i}\rangle\right\vert ^{2}.$$ Moreover, the constant 1/4 is sharp. This is used to prove a refinement of the Grüss inequality.
[Constantin Niculescu (Craiova)]
MSC 2000:
*26D15 Inequalities for sums, series and integrals of real functions
46C05 Geometry and topology of inner product spaces

Keywords: Bessel inequality; Grüss type inequality; inner product space

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