Continuous Inequalities: Introduction, Examples and Related Topics

Classical inequalities in a measure space are usually described by involving finitely many functions \(f_1, \ldots ,f_N\) , \(N=2,3,\ldots \) . In this paper we give an introduction and several examples when such inequalities can be given with infinitely many functions \(f_s\) involved, where index s can even be taken from another measure space. Such a development was also inspired when developing an interpolation between families (continuously many) of Banach spaces.