International Journal of Modern Computation, Information and Communication Technology

ISSN 2581-5954

March-April 2021, Vol. 4, Issue 3-4, p. 13-17.​​

On Isometrically Equivalent Norms in Banach Spaces
P. W.  Mulongo¹, S. Aywa¹, N. B. Okelo²*
¹Department of Mathematics, Kibabii University, Kenya.

²Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and Technology, P. O. Box 210-40601, Bondo-Kenya.
*Corresponding author’s e-mail:


Many studies on equivalent norms are playing an increasingly important role in the theory of Banach spaces. Also in consideration has been the characterization of operator ideals. In this paper we develop isometrically equivalent norms in a Banach space. The objectives of the study are to show that there exists Banach spaces which are isometrically isomorphic; develop an unconditional basis in a Banach space and renorm an equivalent norm under which a Banach space will become a Banach lattice and still remain a banach space. Authors showed that there exists Banach spaces which are isometric and isomorphic. Authors further showed that for a normal Banach space which is absolutely continuous, then there exists an unconditional basis in a Banach space.

Keywords: Banach space; Banach lattice; Unconditional basis; Isomorphic.


  1. Bellenot SF. Prevarieties and intertwined completeness of locally convex spaces. Ann. 1975;217:59-67.
  2. Borwein JM, Jimenez- Sevilla, Moreno JP. Antiproximinal norms in Banach spaces. Journal of Approximation Theory 2002;114:57-69 .
  3. Borza S, Marcoci AN, Persson LE. Best constants between equivalent norms in Lorentz sequence spaces, Journal of Function Spaces and Applications. 2012; 2012;713534.
  4. Diestel J, Morris, SA, Saxou SA. Varieties of linear topological spaces. Trans. Amer. Math. Soc. 1972;172:207-230.
  5. Fourie JH. operators factoring compactly through ABK-space. J. Function Spaces  1979;10:15-40.
  6. Deville R, Godfrey G, Zizter, V, Smoothness and Renormings in Banach space, Longman Scientific & Technical,  1993.
  7. Hogbe N, Bornologies and functional analysis, North Holland Publ. Co., Amsterdam. New York. Oxford, 1977.
  8. Horvath J, Horvath J. Topological vector spaces and distributions, vol. 1 addison- Wesley Pub. Co., Massachusetts. Palo. Aho. London. Don mills Ontario, 1966.
  9. Jarchow H. Nuclear locally convex spaces, Lecture notes 13 Dept. of mathematics, University of Maryland. 1976.
  10. Jarosz K. Any Banach space has an equivalent norm with trivial isometries. Israel Journal of Math. 1988;64:49-56.
  11. Kelvin M. Positive operators on Banach lattice, Lecture notes Series, Department of Mathematics, Kibabii University, Kibabii 2012.
  12. Kothe G. Topological Vector Spaces I, Springer Verlag. Berlin, 1969.
  13. Lindenstrauss J, Tzafriril H. Classsical banach spaces I, Springer Verlag, New York, 1977.
  14. Matt D. Introduction to Bases in Banach spaces.¿pubs2005
  15. Naumann J, Berlin CG, Simader B. A second look on definition and equivalent norms of sobolev spaces, Mathematica Bohemica 1999;23:315-28.
  16. Panchapagesan TV. Unitary operators in Banach spaces. Pacific Journal of Math. 1967;22:465-475.
  17. Pietsch A. Operator ideals, Academic verlag, Berlin, 1978.
  18. Randtke D. Characterization of precompact maps, Schwartz Spaces And Nuclear Spaces. Trans. Amer. Math. Soc. 1972;165:87-101.
  19. Randtke D. A simple example of universal Schwartz space. Proc. Amer. Math. Soc. 1973;37:185-8 .
  20. Ruckle W. On the construction of sequence spaces that have Schauder bases. Canad. J. Math. 1966;18:1281-93.
  21. Ruckle W. Symmetric coordinate spaces and symmetric bases. Canad. J. Math. 1967;19:828-38.
  22. Schaefer H.H. Topological vector spaces, Singer-verlag. Newyork. Heidelberg. Berlin, 1971.
  23. Singer I. Bases I Banach spaces, I Springer verlag Berlin. Heidelberg. New York, 1970.
  24. Smith RJ. Guenhage compacta and Strictly convex dual norms. J. Math. Anal. Appl. 2009;350:745-57 .
  25. Swart J. operator ideals on the class of locall convex spaces. Technical report FA 13 dept of mathematics, potchefatroom University, 1977.
  26. Walter S. Norm attaining operators and renormings of Banach spaces. Israel Journal of Mathematics 1983;44:201-12.
  27. Wilansky A., Functional Analysis, Blaisdel publ. co., New York. Toronto London, 1964.