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

Keiichi Miyajima; Artur Korniłowicz; Yasunari Shidama

doi:10.2478/v10037-012-0011-3

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

Formalized Mathematics

Volume 20 (2012): Issue 1 (January 2012)

By: Keiichi Miyajima, Artur Korniłowicz and Yasunari Shidama

Open Access

|Sep 2012

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].

References

Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
Search in Google Scholar
Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
Search in Google Scholar
Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
Search in Google Scholar
Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
Search in Google Scholar
Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.
Search in Google Scholar
Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
Search in Google Scholar
Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
Search in Google Scholar
Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
Search in Google Scholar
Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.
Search in Google Scholar
Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.
Search in Google Scholar
Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.
Search in Google Scholar
Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Darboux's theorem. Formalized Mathematics, 9(1):197-200, 2001.
Search in Google Scholar
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.
Search in Google Scholar
Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.
Search in Google Scholar
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
Search in Google Scholar
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.10.2478/v10037-009-0021-y
Search in Google Scholar
Keiko Narita, Artur Kornilowicz, and Yasunari Shidama. More on the continuity of real functions. Formalized Mathematics, 19(4):233-239, 2011, doi: 10.2478/v10037-011-0032-3.10.2478/v10037-011-0032-3
Search in Google Scholar
Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
Search in Google Scholar
Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.
Search in Google Scholar
Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.
Search in Google Scholar
Murray R. Spiegel. Theory and Problems of Vector Analysis. McGraw-Hill, 1974.
Search in Google Scholar
Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
Search in Google Scholar
Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
Search in Google Scholar
Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
Search in Google Scholar
Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
Search in Google Scholar
Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992.
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/v10037-012-0011-3 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630

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Language: English

Page range: 79 - 86

Published on: Sep 12, 2012

Published by: University of Białystok

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Related subjects:

Mathematics,

Logic and set theory,

Computer sciences,

Artificial intelligence

© 2012 Keiichi Miyajima, Artur Korniłowicz, Yasunari Shidama, published by University of Białystok
This work is licensed under the Creative Commons License.

Volume 20 (2012): Issue 1 (January 2012)