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].
@article{bwmeta1.element.doi-10_2478_v10037-012-0011-3,
author = {Keiichi Miyajima and Artur Korni\l owicz and Yasunari Shidama},
title = {
Riemann Integral of Functions from R into
n
-dimensional Real Normed Space
},
journal = {Formalized Mathematics},
volume = {20},
year = {2012},
pages = {79-86},
zbl = {1276.26026},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10037-012-0011-3}
}
Keiichi Miyajima; Artur Korniłowicz; Yasunari Shidama.
Riemann Integral of Functions from R into
n
-dimensional Real Normed Space
. Formalized Mathematics, Tome 20 (2012) pp. 79-86. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10037-012-0011-3/
Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. | Zbl 06213858
Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.
Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.
Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.
Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.
Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Darboux's theorem. Formalized Mathematics, 9(1):197-200, 2001.
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.
Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.
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] | Zbl 1276.26025
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]
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] | Zbl 1276.26006
Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.
Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.
Murray R. Spiegel. Theory and Problems of Vector Analysis. McGraw-Hill, 1974.
Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992.