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The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

100%

Narita K., Endou N., Shidama Y.

Formalized Mathematics

2013

tom 21

nr 3

185-191

In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10], we also proved some theorems on bounded functions.

Riemann Integral of Functions from ℝ into Real Banach Space

100%

Narita K., Endou N., Shidama Y.

Formalized Mathematics

2013

tom 21

nr 2

145-152

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.

Ograniczanie wyników

2 Formalized Mathematics

2 Endou N.

2 Narita K.

2 Shidama Y.

2 2013

Wyniki wyszukiwania

The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

Riemann Integral of Functions from ℝ into Real Banach Space