A Characterization Of The Inner Product Spaces Involving Trigonometry

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 $x,y\in X$‎.