Riemann Integral of Functions from R into n -dimensional Real Normed Space

Keiichi Miyajima; Artur Korniłowicz; Yasunari Shidama

Formalized Mathematics (2012)

  • Volume: 20, Issue: 1, page 79-86
  • ISSN: 1426-2630

Abstract

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In this article, we define the Riemann integral on functions R into n-dimensional real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to the wider range. Our method refers to the [21].

How to cite

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Keiichi Miyajima, Artur Korniłowicz, and Yasunari Shidama. " Riemann Integral of Functions from R into n -dimensional Real Normed Space ." Formalized Mathematics 20.1 (2012): 79-86. <http://eudml.org/doc/267713>.

@article{KeiichiMiyajima2012,
abstract = {In this article, we define the Riemann integral on functions R into n-dimensional real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to the wider range. Our method refers to the [21].},
author = {Keiichi Miyajima, Artur Korniłowicz, Yasunari Shidama},
journal = {Formalized Mathematics},
language = {eng},
number = {1},
pages = {79-86},
title = { Riemann Integral of Functions from R into n -dimensional Real Normed Space },
url = {http://eudml.org/doc/267713},
volume = {20},
year = {2012},
}

TY - JOUR
AU - Keiichi Miyajima
AU - Artur Korniłowicz
AU - Yasunari Shidama
TI - Riemann Integral of Functions from R into n -dimensional Real Normed Space
JO - Formalized Mathematics
PY - 2012
VL - 20
IS - 1
SP - 79
EP - 86
AB - In this article, we define the Riemann integral on functions R into n-dimensional real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to the wider range. Our method refers to the [21].
LA - eng
UR - http://eudml.org/doc/267713
ER -

References

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  1. Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
  2. Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. 
  3. Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990. 
  4. Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990. 
  5. Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990. 
  6. Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990. 
  7. Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990. 
  8. Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990. 
  9. Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991. 
  10. Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999. 
  11. Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005. 
  12. Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Darboux's theorem. Formalized Mathematics, 9(1):197-200, 2001. 
  13. Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001. 
  14. Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001. 
  15. 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.[Crossref] Zbl1276.26025
  16. Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rn. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.[Crossref] 
  17. Keiko Narita, Artur Kornilowicz, and Yasunari Shidama. More on the continuity of real functions. Formalized Mathematics, 19(4):233-239, 2011, doi: 10.2478/v10037-011-0032-3.[Crossref] Zbl1276.26006
  18. Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990. 
  19. Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991. 
  20. Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990. 
  21. Murray R. Spiegel. Theory and Problems of Vector Analysis. McGraw-Hill, 1974. 
  22. Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990. 
  23. Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. 
  24. Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990. 
  25. Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990. 
  26. Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992. 

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