# Riemann Integral of Functions from ℝ into Real Banach Space

Keiko Narita; Noboru Endou; Yasunari Shidama

Formalized Mathematics (2013)

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

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topKeiko 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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