Zoltán Boros
Árpád Száz

A weak Schwarz inequality for semi-inner products on groupoids

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 71, 28 - 40 (2018)

MSC: 20L05   Groupoids (i.e. small categories in which all morphisms are isomorphisms)
  46C50   Generalizations of inner products (semi-inner products, partial inner products, etc.)
  39B32   Equations for complex functions
  39B62   Functional inequalities, including subadditivity, convexity, etc.

Abstract:   By introducing appropriate notions of semi-inner products and their induced generalized seminorms on groupoids, we shall prove a weak form of the famous Schwarz inequality. In case of groups, this will be sufficient to prove the subadditivity of the induced generalized seminorms. Thus, some of the results of the theory of inner product spaces can be extended to inner product groups. However, in the near future, we shall only be interested in the corresponding extensions of some fundamental theorems of Gy. Maksa, P. Volkmann, A. Gilányi, J. Rätz and W. Fechner on additive and quadratic functions.

Keywords:   Groupoids, additive functions, semi-inner products, generalized seminorms, Schwarz inequality, triangle inequality.
Notes:   The work of the authors has been supported by the Hungarian Scientific Research Fund (OTKA) Grant K-111651.

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