A Revisit to n-Normed Spaces Through Its Quotient Spaces

Abstract

In this paper, we define several types of continuous mapping in $n$-normed spaces with respect to the norms of its quotient spaces. Then, we show that all types of the continuity are equivalent. We also study contractive mappings on $n$-normed spaces using these norms. In particular, we prove a fixed point theorem for contractive mappings on a closed and bounded set in the $n$-normed space with respect to the norms of its quotient spaces.In the last section we prove a fixed point theorem and give some remarks on the $p$-summable sequence space as an $n$-normed space.