Abstract
In this paper, we explore some geometrical inequalities. We present versions for inner product spaces and we prove that this inequalities can characterize the inner product spaces.
Highlights
It is known that any inner product space is a normed space, but the reverse is not necessary true
In the second section we recall some results included in the references [2] or [15] and we present and prove similar version for the inner product space (Theorem 2.1, 2.2 and 2.3)
The third section is reserved for new characterizations of the inner product space (Theorem 3.1, 3.2 and 3.3)
Summary
It is known that any inner product space is a normed space, but the reverse is not necessary true. Transforming Jordan’s result, a new idea appears: the characterizations of inner product space in terms of inequalities. In this direction, the first known result, due to Schoenberg [14], is contained in the following proposition. The references [7] or [13] contain some recent examples In this context, the aim of our paper is to present new characterizations of the inner product space involving inequalities. The third section is reserved for new characterizations of the inner product space (Theorem 3.1, 3.2 and 3.3) These theorems shows that some results from elementary mathematics are more important than they seem at first view
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