## 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.)

### Keywords:

fixed point theorem; $$n$$-normed space

### Citations:

Zbl 0673.46012; Zbl 0142.39803
Full Text:

### References:

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