Have a personal or library account? Click to login
Riemann Integral of Functions from ℝ into Real Banach Space Cover

Riemann Integral of Functions from ℝ into Real Banach Space

Open Access
|Jun 2013

References

  1. [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
  2. [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
  3. [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
  4. [4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
  5. [5] Józef Białas. Properties of the intervals of real numbers. Formalized Mathematics, 3(2): 263-269, 1992.
  6. [6] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
  7. [7] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
  8. [8] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.
  9. [9] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
  10. [10] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
  11. [11] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
  12. [12] Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.
  13. [13] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
  14. [14] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
  15. [15] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.
  16. [16] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.
  17. [17] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Darboux’s theorem. Formalized Mathematics, 9(1):197-200, 2001.
  18. [18] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability forpartial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001.
  19. [19] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
  20. [20] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.
  21. [21] Keiichi Miyajima, Takahiro Kato, and Yasunari Shidama. Riemann integral of functions from R into real normed space. Formalized Mathematics, 19(1):17-22, 2011. doi:10.2478/v10037-011-0003-8.10.2478/v10037-011-0003-8
  22. [22] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.
  23. [23] Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. More on continuous functions on normed linear spaces. Formalized Mathematics, 19(1):45-49, 2011. doi:10.2478/v10037-011-0008-3.10.2478/v10037-011-0008-3
  24. [24] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.
  25. [25] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.
  26. [26] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.
  27. [27] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.
  28. [28] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
  29. [29] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.
  30. [30] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
  31. [31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
  32. [32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
  33. [33] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
DOI: https://doi.org/10.2478/forma-2013-0016 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630
Language: English
Page range: 145 - 152
Published on: Jun 1, 2013
In partnership with: Paradigm Publishing Services
Publication frequency: 1 issue per year

© 2013 Keiko Narita, Noboru Endou, Yasunari Shidama, published by University of Białystok
This work is licensed under the Creative Commons License.

Volume 21 (2013): Issue 2 (June 2013)