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Bounded Multilinear Functionals and Multicontinuous Functions on n-Normed Spaces

Harmanus Batkunde, Muh. Nur, Al Azhary Masta, Meilin Imelda Tilukay

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.

Bounded Multilinear Functionals and Multicontinuous Functions on n-Normed Spaces

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.
Paper Structure (7 sections, 5 theorems, 63 equations)

This paper contains 7 sections, 5 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Bounded $k$-Linear Functionals (of 1st index).
  4. Bounded $k$-Linear Functionals (of $p$-th index).
  5. Duality Properties for $\mathbf{p=2}$.
  6. Multicontinuous Function on the $n$-Normed Spaces.
  7. Conclusions

Key Result

Lemma 1

The norm in (norm11) is identical with where $\{j_{2},\dots,j_{n}\} \subset \{1,\dots,n\}$.

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • proof
  • Lemma 2
  • proof
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Corollary 1
  • ...and 4 more