Inequalities That Imply the Norm of a Linear Space Is Induced by an Inner Product

The aim of this paper is to investigate when a linear normed space is an inner product space. Several conditions in a linear normed space are formulated with the help of inequalities. Some of them are from the literature and others are new. We prove that these conditions are equivalent with the fact that the norm is induced by an inner product. One of the new results is the following: in an inner product space, the sum of opposite edges of a tetrahedron are the sides of an acute angled triangle. The converse of this result holds also. More precisely, this property characterizes inner product spaces. Another new result is the following: in a tetrahedron, the sum of squares of opposite edges are the lengths of a triangle. We prove also that this property characterizes inner product spaces. In addition, we give simpler proofs to some theorems already known from the publications of other authors.