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Keiichi Miyajima and Takahiro Kato

The Sum and Product of Finite Sequences of Complex Numbers

This article extends the [10]. We define the sum and the product of the sequence of complex numbers, and formalize these theorems. Our method refers to the [11].

Open access

Keiichi Miyajima and Yasunari Shidama

Riemann Integral of Functions from R into Rn

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].

Open access

Noboru Endou, Yasunari Shidama and Keiichi Miyajima

Partial Differentiation on Normed Linear Spaces Rn

Summary. In this article, we define the partial differentiation of functions of real variable and prove the linearity of this operator [18].

Open access

Keiichi Miyajima, Artur Korniłowicz and Yasunari Shidama

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

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].

Open access

Keiichi Miyajima, Artur Korniłowicz and Yasunari Shidama

Summary

In this article, we described the contracting mapping on normed linear space. Furthermore, we applied that mapping to ordinary differential equations on real normed space. Our method is based on the one presented by Schwarz [29].

Open access

Keiichi Miyajima, Takahiro Kato and Yasunari Shidama

Riemann Integral of Functions R into C

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].

Open access

Noboru Endou, Yasunari Shidama and Keiichi Miyajima

The Product Space of Real Normed Spaces and its Properties

In this article, we define the product space of real linear spaces and real normed spaces. We also describe properties of these spaces.

Open access

Keiichi Miyajima, Takahiro Kato and Yasunari Shidama

Riemann Integral of Functions from R into Real Normed Space

In this article, we define the Riemann integral on functions from R into real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to a wider range of functions. The proof method follows the [16].