Bounded Multilinear Functionals and Multicontinuous Functions on n-Normed Spaces

TL;DR

This paper explores bounded multilinear functionals and multicontinuous functions in n-normed spaces, establishing their equivalence, dual space construction, and norm properties, with examples illustrating their application.

Contribution

It introduces a unified framework for boundedness and continuity in n-normed spaces, proving their equivalence and analyzing dual space structures.

Findings

All types of boundedness are equivalent for multilinear functionals.

Dual spaces constructed under different boundedness notions are identical as sets.

Norms on dual spaces are shown to be equivalent.

Abstract

In this paper, we introduced some notions on the n-Normed Spaces. Those are bounded k-linear (or multilinear) functionals and k-continuous (or multicontinuous) functions with k \in \mathbb{N}. We defined k-linear functionals under several types of boundedness, and constructed the corresponding dual spaces based on each type of boundedness. We then proved that these types of boundedness are actually equivalent. This means the boundedness of a multilinear functional can be verified using any of the equivalent notions of boundedness that we defined earlier. The equivalent also implies that all of the resulting dual spaces are identical as a set. We also defined two norms on the dual spaces and showed that both norms are equivalent. Moreover, we gave some examples of bounded k-linear functionals on an n-normed space and calculated their norms with respect to the types of boundedness. We also defined a new notion of k-continuous function in n-normed spaces. Then we gave a relation between the bounded k-linear functional and k-continuous function in n-normed spaces.