|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.|
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
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.