On a generalization of the sine function

Abstract

Abstract. A function s(x, y) = inf λ∈R x+λy x strictly connected with Birkhoff-James orthogonality is considered. This function may be used in functional equations theory, to provide unconditional equations in place of orthogonal equations in the sense of Birkhoff-James. Moreover, we deal with another generalization of the sine function which, in particular leads to a characterization of inner product spaces.

The generalized sine function and geometrical properties of normed spaces

“…Similarly as before, a lot of properties of this function are to be found in [8]. The most important result contained in that paper is the following theorem.…”

“…Using this theorem, we are going to present a characterization of inner product spaces. But first let us quote another result from [8].…”

“…Some further properties of this function are pointed out in [8]; let us only note that it can easily be proved that for all x, y ∈ X and for all α, β ∈ R \ {0} we have s(x, y) = s(αx, βy).…”

“…Let t ∈ 0, π 2 and α 0 be such that φ x,y (t) = α 0 . In the paper [8] (Lemma 2.13) it was proved that the function ψ α0 is concave. Now from Lemma 2.7 we know that the line l = {(a, b) : b = (tan t) a} supports the graph of ψ α0 .…”

“…Now we are going to deal with strict convexity. However, first we need the following lemma which may be viewed as a reformulation of Lemma 2.16 from [8].…”