aequationes mathematicae

, Volume 60, Issue 3, pp 268–282

# Orthogonalities in linear spaces and difference operators

## Summary.

Quite recently C. Alsina, P. Cruells and M. S. Tomás [2], motivated by F. Suzuki's property of isosceles trapezoids, have proposed the following orthogonality relation in a real normed linear space $$(X, \Vert \cdot \Vert)$$: two vectors $$x,y \in X$$ are T-orthogonal whenever¶$$\Vert z-x \Vert^2 + \Vert z-y \Vert^2 = \Vert z \Vert^2 + \Vert z-(x+y) \Vert^2$$¶for every $$z \in X$$. A natural question arises whether an analogue of T-orthogonality may be defined in any real linear space (without a norm structure). Our proposal reads as follows. Given a functional $$\varphi$$ on a real linear space X we say that two vectors $$x,y \in X$$ are $$\varphi$$-orthogonal (and write $$x\perp_{\varphi}y$$) provided that $$\Delta_{x,y}\varphi = 0$$ ($$\Delta_{h_1,h_2}$$ stands here and in the sequel for the superposition $$\Delta_{h_1} \circ \Delta_{h_2}$$ of the usual difference operators).¶We are looking for necessary and/or sufficient conditions upon the functional $$\varphi$$ to generate a $$\varphi$$-orthogonality such that the pair $$X,\perp_{\varphi}$$ forms an orthogonality space in the sense of J. Rätz (cf. [6]). Two new characterizations of inner product spaces as well as a generalization of some results obtained in [2] are presented.

Keywords. Orthogonality relation, Rätz orthogonality space, inner product space, characterization.