The norms on Cartesian powers and sequence spaces

A new family of norms is defined on the Cartesian product of n copies of a given normed space. The new norms are related to the hypergeometric means but are not restricted to the positive real numbers. Quantitative comparisons with the usual p-norms are given. The reflexivity, convexity and smoothness of the norms are shown to be closely related to the corresponding property of the underlying space. Using a limit of isometric embeddings, the norms are extended to spaces of bounded sequences that include all summable sequences. Examples are given to show that the new sequence spaces have very different properties than the usual spaces of p-summable sequences.