# Riemann Integral of Functions from R into R n

Keiichi Miyajima; Yasunari Shidama

Formalized Mathematics (2009)

- Volume: 17, Issue: 2, page 179-185
- ISSN: 1426-2630

## Access Full Article

top## Abstract

top## How to cite

topKeiichi Miyajima, and Yasunari Shidama. " Riemann Integral of Functions from R into R n ." Formalized Mathematics 17.2 (2009): 179-185. <http://eudml.org/doc/267222>.

@article{KeiichiMiyajima2009,

abstract = {In this article, we define the Riemann Integral of functions from R into Rn, and prove the linearity of this operator. The presented method is based on [21].},

author = {Keiichi Miyajima, Yasunari Shidama},

journal = {Formalized Mathematics},

language = {eng},

number = {2},

pages = {179-185},

title = { Riemann Integral of Functions from R into R n },

url = {http://eudml.org/doc/267222},

volume = {17},

year = {2009},

}

TY - JOUR

AU - Keiichi Miyajima

AU - Yasunari Shidama

TI - Riemann Integral of Functions from R into R n

JO - Formalized Mathematics

PY - 2009

VL - 17

IS - 2

SP - 179

EP - 185

AB - In this article, we define the Riemann Integral of functions from R into Rn, and prove the linearity of this operator. The presented method is based on [21].

LA - eng

UR - http://eudml.org/doc/267222

ER -

## References

top- [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
- [2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
- [3] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990.
- [4] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.
- [5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
- [6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
- [7] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
- [8] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 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, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.[Crossref]
- [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] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.
- [16] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.
- [17] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.
- [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] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- [23] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
- [24] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

## Citations in EuDML Documents

top- Keiichi Miyajima, Takahiro Kato, Yasunari Shidama, Riemann Integral of Functions R into C
- Keiichi Miyajima, Artur Korniłowicz, Yasunari Shidama, Riemann Integral of Functions from R into n -dimensional Real Normed Space
- Keiko Narita, Artur Kornilowicz, Yasunari Shidama, More on the Continuity of Real Functions
- Takao Inoué, Noboru Endou, Yasunari Shidama, Differentiation of Vector-Valued Functions on n -Dimensional Real Normed Linear Spaces
- Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama, Higher-Order Partial Differentiation
- Keiko Narita, Artur Korniłowicz, Yasunari Shidama, The Differentiable Functions from R into R n
- Keiichi Miyajima, Artur Korniłowicz, Yasunari Shidama, Contracting Mapping on Normed Linear Space

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.