aequationes mathematicae

, Volume 60, Issue 3, pp 268–282 | Cite as

Orthogonalities in linear spaces and difference operators


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. 


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Copyright information

© Birkhäuser Verlag, Basel, 2000

Authors and Affiliations

  • R. Ger
    • 1
  1. 1. Institute of Mathematics, Silesian University, Bankowa 14, PL-40-007 Katowice, Poland, e-mail:

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