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The generalized sine function and geometrical properties of normed spaces

Szostok T.

Opuscula Mathematica

2015

Vol. 35, no. 1

117--126

Let [formula] be a nornied space. We deal here with a function s : X x X —> R given by the formula [formula] (for x = 0 we must define it separately). Then we take two unit vectors x and y such that y is orthogonal to x in the Birkhoff-James sense. Using these vectors we construct new functions Φx,y which are defined on R. If X is an inner product space, then Φx, y = sin and, therefore, one may call this function a generalization of the sine function. We show that the properties of this function are connected with geometrical properties of the normed space X.

A comparison of some conditional functional equations

Chmieliński J.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2005

Vol. 45, [Z] 2

137--144

For real inner product spaces we consider and compare the classes of mappings satisfying some conditional and unconditional functional equations.