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The space of \(p\)-summable sequences and its natural \(n\)-norm. (English) Zbl 1002.46007
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 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 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.

MSC:
46A70 Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
46B45 Banach sequence spaces
46B99 Normed linear spaces and Banach spaces; Banach lattices
47H10 Fixed-point theorems
46B20 Geometry and structure of normed linear spaces
46A19 Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.)
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