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Keiichi Miyajima, Takahiro Kato, Yasunari Shidama (2010)
Formalized Mathematics
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In this article, we define the Riemann Integral on functions R into C and proof the linearity of this operator. Especially, the Riemann integral of complex functions is constituted by the redefinition about the Riemann sum of complex numbers. Our method refers to the [19].
Keiichi Miyajima, Yasunari Shidama (2009)
Formalized Mathematics
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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].
Keiichi Miyajima, Artur Korniłowicz, Yasunari Shidama (2012)
Formalized Mathematics
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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].
Keiko Narita, Noboru Endou, Yasunari Shidama (2013)
Formalized Mathematics
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In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the...
Tyszkowska, Ewa (2005)
Beiträge zur Algebra und Geometrie
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Mendoza Torress, F.J. (2010)
Annals of Functional Analysis (AFA) [electronic only]
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Hiroshi Yamazaki, Yasunari Shidama, Yatsuka Nakamura, Chanapat Pacharapokin (2009)
Formalized Mathematics
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In this article we prove Cauchy-Riemann differential equations of complex functions. These theorems give necessary and sufficient condition for differentiable function.
B. Bongiorno, Washek Frank Pfeffer (1992)
Commentationes Mathematicae Universitatis Carolinae
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We present a descriptive definition of a multidimensional generalized Riemann integral based on a concept of generalized absolute continuity for additive functions of sets of bounded variation.
Keiko Narita, Noboru Endou, Yasunari Shidama (2013)
Formalized Mathematics
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In this article we deal with the Riemann integral of functions from R into a real Banach space. The last theorem establishes the integrability of continuous functions on the closed interval of reals. To prove the integrability we defined uniform continuity for functions from R into a real normed space, and proved related theorems. We also stated some properties of finite sequences of elements of a real normed space and finite sequences of real numbers. In addition we proved some theorems...