A note on Euclidean spaces $\mathbb{R}^n$ and  $n$-normed spaces

Mochammad Idris

doi:10.62918/hjma.v1i2.9

Authors

Mochammad Idris Universitas Lambung Mangkurat

DOI:

https://doi.org/10.62918/hjma.v1i2.9

Abstract

This article gives us a relation between Euclidean space $\mathbb{R}^n$ and a subspace of $X$ (an $n$-normed space) using properties of determinant of square matrices. Some functionals will be investigated on these spaces. Furthermore, we obtain definition of inner product and $m$-inner product with $m<n$ on the subspace of $X$.

References

S. Ekariani, H. Gunawan, M. Idris, A contractive mapping theorem on the $n$-normed space of $p$-summable sequences, J. Math. Analysis 4-1, (2013), 1-7.

C. R. Diminnie, S. Gahler, A. White, $2$-inner product spaces, Demonstratio Math. 6, (1973), 525-536.

S. Gahler, Lineare $2$-normietre Raume, Math. Nachr. 28, (1965), 1-43.

S. Gahler, Uber $2$-Banach Ra}ume, Math. Nachr. 42, (1969), 335-347.

H. Gunawan, An inner product that makes a set of vectors orthonormal, Austral. Math. Soc. Gaz. 28, (2001), 194-197.

H. Gunawan, The space of $p$-summable sequences and its natural $n$-norms, Bull. Austral. Math. Soc. 64, (2001), 137-147.

H. Gunawan, On $n$-inner products, $n$-norms, and the Cauchy-Schwarz inequality, Sci. Math. Jpn. 55, (2002), 53-60.

H. Gunawan, O. Neswan, W. Setya-Budhi, A formula for angles between two subspaces of inner product spaces, Beitr. Algebra Geom. 46, (2005), 311-329.

S. Konca, M. Idris, H. Gunawan, $p$-summable sequence spaces with inner products, Beu J. Sci. Techn. 5-1, (2015), 37-41.

S. Konca, M. Idris, Equivalence among three 2-norms on the space of $p$-summable sequences, Journal of Inequalities and Special Functions 7-4, (2015), 218-224.

E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, Inc., New York, 1978.

A. Misiak, $n$-inner product spaces, Math. Nachr. 140, (1989), 299-319.

A. Misiak, Orthogonality and orthonormality in $n$-inner product spaces, Math. Nachr. 143, (1989) 249-261.