Some characterizations of inner product spaces based on angle

A problem in functional analysis that arises naturally is about finding necessary and sufficient conditions for a normed space to be an inner product space. By answering this question, mathematicians try to understand the inner product and normed spaces features. In this note, we have discussed this issue and we prove some results concerned with it. We introduce a notion of angle between two vectors in a normed space, denoted by $A_\theta(.,.)$ where $\theta\neq{k\pi\over2}$. We also speak about a notion of orthogonality concerning it, we call it $\theta$-orthogonality.