Ubiquitously Continuous on Seminorm
Abstract
The term ubiquitously continuous is defined with the purpose of investigating the continuity of a linear operator based on the presence of normed spaces infinite dimensional linear subspaces. Norms and seminorms are parts of the study of normed spaces and obviously are both able to be continuous or discontinuous with respect to their spaces. Hence, this paper aims to explore how ubiquitously continuity of norms and seminorms with respect to infinite dimensional vector spaces. By using available theorems and lemmas, norms and seminorms are generally ubiquitously continuous unless the norms and seminorms themselves are discontinuous at a certain point.
References
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[2] E. Kreyszig, Introductory functional analysis with applications. 1978.
[3] H. Anton and C. Rorres, “Elementary Linear Algebra Applications,” 2014.
[4] B. Rynne and M. Youngson, “Linear Functional Analysis,” Swiss, 2008.
[5] W. Rudin, Functional analysis. 1991.
[6] D. R. Wilkins, “Normed Vector Spaces and Functional Analysis,” 1997.
[7] I. Wilde, “Functional Analysis ‘Topological Vector Spaces’ version,” 2003.
[8] M. A. Akcocglu, P. F. A. Bartha, and D. Minh Ha, “Analysis in Vector Spaces,” 2009.
[9] Mehdi, “Continuity of Seminorm on Topological Vector Spaces,” Studia Mathematica, 1959.
[10]M. Goldberg, “Continuity of seminorms on finite-dimensional vector spaces,” Linear Algebra Appl, vol. 515, pp. 175–179, Feb. 2017, doi: 10.1016/j.laa.2016.11.013.
[11]R. W. Cross, “Some continuity properties of linear transformations in normed spaces,” Glasgow Mathematical Journal, vol. 30, no. 2, pp. 243–247, 1988, doi: 10.1017/S0017089500007291.
[12]J. Chmieliński and M. Goldberg, “Continuity and discontinuity of seminorms on infinite-dimensional vector spaces. II,” Linear Algebra Appl, vol. 594, pp. 249–261, Jun. 2020, doi: 10.1016/j.laa.2020.02.023.
[13]J. H. Loxton and W. O. Neumann, “Introduction to the Analysis of Normed Linear Spaces,” 2000.
[14]B. Beuzamy, “Introduction to Banach Spaces and Their Geometry”. 1982.
[15]E. M. Stein and R. Shakarchi, “Functional Analysis ‘Introduction to Further Topics in Analysis’”. 2011.
[16]J. Chmieliński and M. Goldberg, “Continuity and discontinuity of seminorms on infinite-dimensional vector spaces,” Linear Algebra Appl, vol. 578, pp. 153-158, Jun. 2020.
Published
2024-12-31
How to Cite
ROSMAYA, Dwi Ajeng et al.
Ubiquitously Continuous on Seminorm.
Proceedings of the International Conference on Green Technology, [S.l.], v. 14, n. 1, dec. 2024.
ISSN 2580-7099.
Available at: <https://conferences.uin-malang.ac.id/index.php/ICGT/article/view/3212>. Date accessed: 07 mar. 2026.
Section
Pure and Applied Mathematics
