Abstract
In this paper we will give a new characterization of the inner product space which use the trigonometry. We conclude that a normed space $\left( X,\left\Vert \cdot \right\Vert \right)$ is an inner product space if and only if there exists $\alpha\in\mathbb{R}\backslash \pi\mathbb{Q}$ so that $$\left\Vert x\cos\alpha + y\sin\alpha\right\Vert^{2}+\left\Vert y\cos\alpha - x\sin\alpha\right\Vert^{2}=\left\Vert x\right\Vert^{2}+\left\Vert y\right\Vert^{2},$$ for any $x,y\in X$.
Citation
Mihai Monea. Mihai Opincariu. Marian Stroe. Dan Ştefan Marinescu. "A Characterization Of The Inner Product Spaces Involving Trigonometry." Ann. Funct. Anal. 4 (1) 109 - 113, 2013. https://doi.org/10.15352/afa/1399899840
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