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2008 | 16 | 4 | 361-369
Tytuł artykułu

OnL1Space Formed by Real-Valued Partial Functions

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This article contains some definitions and properties refering to function spaces formed by partial functions defined over a measurable space. We formalized a function space, the so-called L1 space and proved that the space turns out to be a normed space. The formalization of a real function space was given in [16]. The set of all function forms additive group. Here addition is defined by point-wise addition of two functions. However it is not true for partial functions. The set of partial functions does not form an additive group due to lack of right zeroed condition. Therefore, firstly we introduced a kind of a quasi-linear space, then, we introduced the definition of an equivalent relation of two functions which are almost everywhere equal (=a.e.), thirdly we formalized a linear space by taking the quotient of a quasi-linear space by the relation (=a.e.).MML identifier: LPSPACE1, version: 7.9.03 4.108.1028
Słowa kluczowe
Wydawca
Rocznik
Tom
16
Numer
4
Strony
361-369
Opis fizyczny
Daty
wydano
2008-01-01
online
2009-03-20
Twórcy
  • Shinshu University, Nagano, Japan
autor
  • Gifu National College of Technology, Japan
  • Shinshu University, Nagano, Japan
Bibliografia
  • [1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565-582, 2001.
  • [2] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.
  • [3] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.
  • [4] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163-171, 1991.
  • [5] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.
  • [6] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.
  • [7] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
  • [8] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
  • [9] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
  • [10] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
  • [11] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
  • [12] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006.
  • [13] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1)35-40, 1990.
  • [14] Jarosław Kotowicz and Yuji Sakai. Properties of partial functions from a domain to the set of real numbers. Formalized Mathematics, 3(2):279-288, 1992.
  • [15] Andrzej Nedzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.
  • [16] Henryk Oryszczyszyn and Krzysztof Prżmowski. Real functions spaces. Formalized Mathematics, 1(3):555-561, 1990.
  • [17] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
  • [18] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.
  • [19] Yasunari Shidama and Noboru Endou. Integral of real-valued measurable function.Formalized Mathematics, 14(4):143-152, 2006.
  • [20] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.
  • [21] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.
  • [22] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297-301, 1990.
  • [23] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
  • [24] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
  • [25] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_v10037-008-0044-9
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