Three Equivalent n-Norms on the Space of p-Summable Sequences

Year 2019, Volume: 2 Issue: 2, 123 - 129, 20.12.2019

Muh Nur , Hendra Gunawan

https://doi.org/10.33401/fujma.635754

Cited By: 1

Abstract

Given a normed space, one can define a new $n$-norm using a semi-inner product $g$ on the space, different from the $n$-norm defined by G\"{a}hler. In this paper, we are interested in the new $n$-norm which is defined using such a functional $g$ on the space $\ell^p$ of $p$-summable sequences, where $1\le p<\infty$. We prove particularly that the new $n$-norm is equivalent with the one defined previously by Gunawan on $\ell^p$.

Keywords

Equivalence, n-norm, Semi-inner product g, p-summable sequence space

Supporting Institution

ITB Research and Innovation Program 2019

References

• [1] J. R. Giles, Classes of semi-inner product spaces, Trans. Amer. Math. Soc., 129–3 (1967), 436–446.
• [2] S. S. Dragomir, Semi-inner Products and Applications, Nova Science Publishers, 2004.
• [3] P. M. Milicic, Sur le g-angle dans un espace norme, Mat. Vesnik, 45 (1993), 43–48.
• [4] M. Nur, H. Gunawan, A new orthogonality and angle in a normed space, Aequationes Math., 93 (2019), 547–555.
• [5] M. Nur, H.Gunawan, O. Neswan, A formula for the g-angle between two subspaces of a normed space, Beitr. Algebra Geom., 59-1 (2018), 133–143.
• [6] M. Nur, H. Gunawan, A note on the formula of the g-angle between two subspaces in a normed space, (Feb. 2019) arXiv:1809.01909v2 [math.FA].
• [7] S. Gahler, Lineare 2-normierte raume, Math. Nachr., 28 (1964), 1–43.
• [8] S. Gahler, Untersuchungen über verallgemeinerte m-metrische raume. I”, Math. Nachr., 40 (1969), 165–189.
• [9] S. Gahler, Untersuchungen über verallgemeinerte m-metrische raume. II”, Math. Nachr., 40 (1969), 229–264.
• [10] S. Gahler, Untersuchungen über verallgemeinerte m-metrische raume. III, Math. Nachr., 41 (1970), 23–26.
• [11] S. Ekariani, H. Gunawan, M. Idris, A contractive mapping theorem on the n-normed space of p-summable, J. Math. Anal., 41 (2013), 1–7.
• [12] S. M. Gozali, H. Gunawan, O. Neswan, On n-norms and bounded n-linear functionals in a Hilbert space, Ann. Funct. Anal., 1 (2010), 72–79.
• [13] H. Gunawan, On n-inner products, n-norms, and the Cauchy-Schwarz inequality, Sci. Math. Japon., 55 (2002), 53–60.
• [14] H. Gunawan, H. Mashadi, On n-normed spaces, Int. J. Math. Math. Sci. 27 (2001), 631–639.
• [15] H. Gunawan, W. Setya-Budhi, M. Mashadi, S.Gemawati, On volumes of n-dimensional parallelepipeds in `p spaces, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat., 16 (2005), 48–54.
• [16] H. Gunawan, The space of p-summable sequences and its natural n-norms, Bull. Austral. Math. Soc., 64 (2001), 137–147.
• [17] Ş. Konca, M. Idris, Equivalence among three 2-norms on the space of p-summable sequences, J. Inequal. Spec. Funct., 7(4), (2016) 218–224.
• [18] A. Mutaqin, H. Gunawan, Equivalence of n-norms on the space of p-summable sequences, J. Indones. Math. Soc., 16 (2010), 39–49.
• [19] R. A. Wibawa-Kusumah, H. Gunawan, Two equivalent n-norms on the space of p-summable sequences, Period. Math. Hungar., 67–1 (2013), 63–69.
• [20] P. M. Milicic, On the Gram-Schmidt projection in normed spaces, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat., 4 (1993), 89–96.
• [21] S. Kurepa, On the Buniakowsky-Cauchy-Schwarz inequality, Glas. Mat. III Ser., 21(1) (1966), 147–158.
• [22] F. R. Gantmacher, The Theory of Matrices, AMS Chelsea Publishing, 1, 2000.