## Riemann Integral of Functions from R into Rn

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

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

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].

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].

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

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].

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

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].

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].