# Riemann Integral of Functions from ℝ into Real Banach Space

Formalized Mathematics (2013)

• Volume: 21, Issue: 2, page 145-152
• ISSN: 1426-2630

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## Abstract

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

## How to cite

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Keiko Narita, Noboru Endou, and Yasunari Shidama. "Riemann Integral of Functions from ℝ into Real Banach Space." Formalized Mathematics 21.2 (2013): 145-152. <http://eudml.org/doc/267180>.

@article{KeikoNarita2013,
abstract = {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.},
author = {Keiko Narita, Noboru Endou, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {formalization of Riemann integral},
language = {eng},
number = {2},
pages = {145-152},
title = {Riemann Integral of Functions from ℝ into Real Banach Space},
url = {http://eudml.org/doc/267180},
volume = {21},
year = {2013},
}

TY - JOUR
AU - Keiko Narita
AU - Noboru Endou
AU - Yasunari Shidama
TI - Riemann Integral of Functions from ℝ into Real Banach Space
JO - Formalized Mathematics
PY - 2013
VL - 21
IS - 2
SP - 145
EP - 152
AB - 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.
LA - eng
KW - formalization of Riemann integral
UR - http://eudml.org/doc/267180
ER -

## References

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