# Riemann Integral of Functions from ℝ into Real Banach Space

Open access

## Summary

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 about the convergence of sequences. We applied definitions introduced in the previous article [21] to the proof of integrability.

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

[3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.

[5] Józef Białas. Properties of the intervals of real numbers. Formalized Mathematics, 3(2): 263-269, 1992.

[6] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.

[7] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.

[8] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.

[9] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.

[10] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

[11] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.

[12] Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.

[13] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

[14] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.

[15] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.

[16] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.

[17] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Darboux’s theorem. Formalized Mathematics, 9(1):197-200, 2001.

[18] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability forpartial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001.

[19] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.

[20] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.

[21] 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.

[22] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.

[23] Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. More on continuous functions on normed linear spaces. Formalized Mathematics, 19(1):45-49, 2011. doi:10.2478/v10037-011-0008-3.

[24] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.

[25] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.

[26] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.

[27] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.

[28] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.

[29] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.

[30] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.

[31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

[32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.

[33] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

# Formalized Mathematics

## (a computer assisted approach)

### Journal Information

SCImago Journal Rank (SJR) 2017: 0.119
Source Normalized Impact per Paper (SNIP) 2017: 0.237

Target Group

researchers in the fields of formal methods and computer-checked mathematics

### Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 87 87 13
PDF Downloads 31 31 4