The space of \(p\)-summable sequences and its natural \(n\)-norm (Q2748274)

Let \(n\) be a nonnegative integer. In this note, the author studies the space \(\ell^p\), \(1 \leq p \leq \infty,\) equipped the natural \(n\)-norm in the sense of \textit{A. Misiak} [Math. Nachr. 140, 299-319 (1989; Zbl 0673.46012)]. The concept of an \(n\)-norm is a generalization of the concept of a \(2\)-norm developed by \textit{S. Gähler} [Math. Nach. 28, 1-43 (1964; Zbl 0142.39803)]. It is shown in the paper that \(\ell^p\), \(1 \leq p \leq \infty,\) is complete with respect to the \(n\)-norm. The author also proves a fixed point theorem for \(\ell^p\) as an \(n\)-normed space.