The \(p\)-\(HH\) norms on Cartesian powers and sequence spaces
Let \(X\) be a Banach space and let \(n\) be a positive integer greater than or equal to \(2\). The way in which \(X^n\) is usually normed consists in calculating the norm of the coordinates of a vector of \(X^n\) and then calculating a mean (usually a \(p\)-norm) of them. This implies that the norm of a vector depends only on the norms of its coordinates, a fact which is useful in some problems but which the authors would like to avoid here. The idea in this paper is to introduce norms on \(X^n\) in a completely different way: for a vector in \(X^n\), a mean using the coordinates is calculated and then the norm of the mean in \(X\) gives the so-called \(p\)-\(HH\) norm of the vector. The case \(n=2\) was previously introduced by the first author of this paper and \textit{S.\,S.\thinspace Dragomir} [Math.\ Inequal.\ Appl.\ 13, No.\,1, 1--32 (2010; Zbl 1183.26025)]. The authors prove that the \(p\)-\(HH\) norms are actually norms in \(X^n\) which are equivalent to the usual \(p\)-norms but not equal and study their convexity and smoothness properties. Using a limit of isometric embedding, the \(p\)-\(HH\) norms are extended to spaces of bounded sequences which include all summable sequences. Examples are given to show that the new sequence spaces have very different properties from the usual spaces of \(p\)-summable sequences.
scientific article; zbMATH DE number 5597553
scientific article