Rings of continuous linear transformations on a topological vector space

Jose, Ajo; Namboothiri, N. M. Madhavan; Thomas, Jineesh

doi:10.47974/JIM-2164

Open Access Research Article

Rings of continuous linear transformations on a topological vector space

^*Ajo JoseCorresponding authorajostpc@gmail.comResearch Center of Mathematics St. Thomas College Palai; Department of Mathematics St. Pius X College RajapuramResearch Center of Mathematics St. Thomas College PalaiKottayam, Kerala, India0000-0001-8429-6088View full profile → , N. M. Madhavan Namboothirimadhavangck@gmail.comGovernment College PoopparaMurikkumthotti, Kerala, 685619, IndiaView full profile → , Jineesh Thomasjineeshthomas@gmail.comDepartment of Mathematics Government Residential Women’s Polytechnic CollegePayyannur, Kerala, 670307, India0000-0002-9175-8348View full profile →

* Corresponding author · click or hover a name for details

pp. 1623–1634Vol. 28Issue 4June 2025DOI: 10.47974/JIM-2164XML

Received:: 13 Mar 2024
Published Online:: 03 Jun 2025
Article type:: Research Article
Language:: EN
Article no.:: JIM-2164
Pages:: 1623–1634

Abstract

A structural analysis of the ring CL(X) of all linear continuous transformations on a topological vector space X is presented here. It explores the algebraic cum topological properties of CL(X) in relation with the structural features of X. Specifically, we discussed the relations between left ideals, zero-sets and z-filters in a topological vector space X and proved that if X is a finite dimensional topological vector space, the left ideal generated by the Zero set of a left ideal I in CL(X) is I itself.

Keywords

Topological vector space Ring Zero set Z-filters Left ideal Weak topology

Subject Classifications

16W8046E2546H3547L2054H1357N17

References

[1] K. Bryan and T. Leise, “The $25,000,000,000 eigenvector: The linear algebra behind Google,” SIAM, vol. 48, no. 3, pp. 569-581 (2006).
[2] C. M. R. Caridade, R. P. Nunes, and Q. D. Nora, “Applying image processing techniques to motivate students in linear algebra classes,” Learning, Citeseer, vol. 2, pp. 114-121 (2011).
[3] F. Colonius and W. Kliemann, Dynamical Systems and Linear Algebra, American Mathematical Society, Providence, RI, vol. 158 (2014).
[4] F. Colonius and W. Kliemann, The Dynamics of Control, Springer Science+Business Media, LLC, pp. 309-497 (2000).
[5] Y. Cui, J. Qu, and G. Zhou, “Application of linear transformation in numerical calculation,” Journal of Chemical and Pharmaceutical Research, vol. 6, no. 3, pp. 170-178 (2014).
[6] L. Gillman and M. Jerison, Rings of Continuous Functions, D. Van Nostrand Company, Princeton (1960).
[7] A. Hladnik, “Image compression and face recognition: Two image processing applications of principal component analysis,” International Circular of Graphic Education and Research, vol. 6, no. 4, pp. 56-61 (2013).
[8] H. Z. Ibrahim, “On strongly α-topological vector spaces,” Journal of Interdisciplinary Mathematics, vol. 25, no. 8, pp. 2539-2551 (2022).
[9] N. J. Laustsen and J. T. White, “Subspaces that can and cannot be the kernel of a bounded operator on a Banach space,” arXiv:1811.02399v3 (2018).
[10] A. N. Langville and C. D. Meyer, “The use of linear algebra by web search engines,” IMAGE Newsletter, vol. 33, no. 4, pp. 2-6 (2004).
[11] D. C. Lay, S. R. Lay, and J. J. McDonald, Linear Algebra and Its Applications, 5th ed., Pearson, pp. 222-223 (2016).
[12] B. V. Limaye, Functional Analysis, 2nd ed., New Age International (P) Ltd (1995).
[13] H. L. Royden, Real Analysis, 3rd ed., Prentice-Hall of India, pp. 234-235 (2006).
[14] W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill Education, pp. 10-11 (2006).
[15] O. Toeplitz, “Ueber die Auflösung unendlichvieler linearer Gleichungen mit unendlichvielen Unbekannten,” Rend. Circolo Mat. Palermo, vol. 28, pp. 88-96 (1909).

Views: 153Downloads: 8Citations: 0