Have a personal or library account? Click to login
Separability of Real Normed Spaces and Its Basic Properties Cover

Separability of Real Normed Spaces and Its Basic Properties

Formalized Mathematics
Volume 23 (2015): Issue 1 (March 2015)
By: Kazuhisa Nakasho and  Noboru Endou  
Open Access
|Mar 2015

References

  1. [1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565–582, 2001.
    Search in Google Scholar
  2. [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.
    Search in Google Scholar
  3. [3] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.
    Search in Google Scholar
  4. [4] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.
    Search in Google Scholar
  5. [5] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.
    Search in Google Scholar
  6. [6] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.
    Search in Google Scholar
  7. [7] Nicolas Bourbaki. Topological vector spaces: Chapters 1-5. Springer, 1981.
    Search in Google Scholar
  8. [8] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.
    Search in Google Scholar
  9. [9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.
    Search in Google Scholar
  10. [10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.
    Search in Google Scholar
  11. [11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.
    Search in Google Scholar
  12. [12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.
    Search in Google Scholar
  13. [13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.
    Search in Google Scholar
  14. [14] N. J. Dunford and T. Schwartz. Linear operators I. Interscience Publ., 1958.
    Search in Google Scholar
  15. [15] Noboru Endou, Yasunari Shidama, and Katsumasa Okamura. Baire’s category theorem and some spaces generated from real normed space. Formalized Mathematics, 14(4): 213–219, 2006. doi:10.2478/v10037-006-0024-x.10.2478/v10037-006-0024-x
    Search in Google Scholar
  16. [16] Andrey Kolmogorov and Sergei Fomin. Elements of the Theory of Functions and Functional Analysis [Two Volumes in One]. Martino Fine Books, 2012.
    Search in Google Scholar
  17. [17] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.
    Search in Google Scholar
  18. [18] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.
    Search in Google Scholar
  19. [19] Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Topological properties of real normed space. Formalized Mathematics, 22(3):209–223, 2014. doi:10.2478/forma-2014-0024.10.2478/forma-2014-0024
    Search in Google Scholar
  20. [20] Keiko Narita, Noboru Endou, and Yasunari Shidama. Dual spaces and Hahn-Banach theorem. Formalized Mathematics, 22(1):69–77, 2014. doi:10.2478/forma-2014-0007.10.2478/forma-2014-0007
    Search in Google Scholar
  21. [21] Keiko Narita, Noboru Endou, and Yasunari Shidama. Bidual spaces and reflexivity of real normed spaces. Formalized Mathematics, 22(4):303–311, 2014. doi:10.2478/forma-2014-0030.10.2478/forma-2014-0030
    Search in Google Scholar
  22. [22] Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29–34, 1993.
    Search in Google Scholar
  23. [23] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.
    Search in Google Scholar
  24. [24] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.
    Search in Google Scholar
  25. [25] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2004.
    Search in Google Scholar
  26. [26] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 1990.
    Search in Google Scholar
  27. [27] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.
    Search in Google Scholar
  28. [28] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.
    Search in Google Scholar
  29. [29] Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581–588, 1990.
    Search in Google Scholar
  30. [30] Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847–850, 1990.
    Search in Google Scholar
  31. [31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.
    Search in Google Scholar
  32. [32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.
    Search in Google Scholar
  33. [33] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.
    Search in Google Scholar
  34. [34] Kosaku Yoshida. Functional Analysis. Springer, 1980.
    Search in Google Scholar
DOI: https://doi.org/10.2478/forma-2015-0005 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630
Journal RSS Feed
Language: English
Page range: 59 - 65
Submitted on: Feb 26, 2015
|
Published on: Mar 31, 2015
Published by: University of Białystok
In partnership with: Paradigm Publishing Services
Publication frequency: 1 issue per year
Keywords:
46B20,
46A19,
03B35
Related subjects:
Mathematics,
General mathematics,
Computer sciences,
Computer sciences, other

© 2015 Kazuhisa Nakasho, Noboru Endou, published by University of Białystok
This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Previous articleVolume 23 (2015): Issue 1 (March 2015)Next article