Some characterizations of inner product spacesbased on angle

Abstract

A problem in functional analysis that arises naturally is about finding necessary andsufficient conditions for a normed space to be an inner product space. By answering thisquestion, 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. Weintroduce a notion of angle between two vectors in a normed space, denoted by$A_θ$(.,.)whereθ̸=kπ2. We also speak about a notion of orthogonality concerning it, we call itθ-orthogonality.