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2013 | 21 | 3 | 185-191
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Wydawca
Rocznik
Tom
21
Numer
3
Strony
185-191
Opis fizyczny
Daty
wydano
2013-10-01
Twórcy
autor
  • Hirosaki-city Aomori, Japan
autor
  • Gifu National College of Technology Japan
  • Shinshu University Nagano, Japan
Bibliografia
  • [1] Józef Białas. Properties of the intervals of real numbers. Formalized Mathematics, 3(2): 263-269, 1992.
  • [2] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
  • [3] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
  • [4] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
  • [5] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
  • [6] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
  • [7] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.
  • [8] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001.
  • [9] 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.[Crossref]
  • [10] Keiichi Miyajima, Artur Korniłowicz, and Yasunari Shidama. Riemann integral of functions from R into n-dimensional real normed space. Formalized Mathematics, 20(1):79-86, 2012. doi:10.2478/v10037-012-0011-3.[Crossref]
  • [11] Keiko Narita, Noboru Endou, and Yasunari Shidama. Riemann integral of functions from R into real Banach space. Formalized Mathematics, 21(2):145-152, 2013. doi:10.2478/forma-2013-0016.[Crossref]
  • [12] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.
  • [13] 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.[Crossref]
  • [14] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.
  • [15] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.
  • [16] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.
  • [17] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.
  • [18] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
  • [19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
  • [20] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
  • [21] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
  • [22] Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_forma-2013-0020
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