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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.

Orthogonal stability of the Cauchy functional equation on balls in normed linear spaces

Sikorska J.

Demonstratio Mathematica

2004

Vol. 37, nr 3

579--596

We study the stability of some functional equations postulated for orthogonal vectors in a ball centered at the origin. The maps considered are denned on a finite-dimensional normed linear space with Birkhoff-James orthogonality and take their values in a real sequentially complete linear topological space. The main results establish the stability of the corresponding conditional Cauchy functional equation on a half-ball and in uniformly convex spaces on a whole ball. The methods used in the first part of the paper are similar to those from [10]. Since, however, now in a general structure, some additional problems arise, we need several new tools.