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2014 | 22 | 1 | 69-77
Tytuł artykułu

Dual Spaces and Hahn-Banach Theorem

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of real linear spaces. In addition, we proved properties of the norm about elements of dual spaces. For the proof we referred to descriptions in the article [21]. Finally, applying theorems of the second section, we proved the Hahn-Banach extension theorem in real normed spaces. We have used extensively used [17].
Słowa kluczowe
Wydawca
Rocznik
Tom
22
Numer
1
Strony
69-77
Opis fizyczny
Twórcy
autor
  • Hirosaki-city Aomori, Japan
autor
  • Gifu National College of Technology Gifu, Japan
  • Shinshu University Nagano, Japan
Bibliografia
  • [1] Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213-225, 1992.
  • [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
  • [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
  • [4] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.
  • [5] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
  • [6] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
  • [7] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
  • [8] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
  • [9] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
  • [10] Noboru Endou, Yasumasa Suzuki, and Yasunari Shidama. Real linear space of real sequences. Formalized Mathematics, 11(3):249-253, 2003.
  • [11] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.
  • [12] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.
  • [13] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.
  • [14] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
  • [15] Anna Justyna Milewska. The Hahn Banach theorem in the vector space over the field of complex numbers. Formalized Mathematics, 9(2):363-371, 2001.
  • [16] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.
  • [17] Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29-34, 1993.
  • [18] Henryk Oryszczyszyn and Krzysztof Prazmowski. Real functions spaces. Formalized Mathematics, 1(3):555-561, 1990.
  • [19] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263-264, 1990.
  • [20] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.
  • [21] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.
  • [22] Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377-380, 2003.
  • [23] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.
  • [24] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.
  • [25] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297-301, 1990.
  • [26] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
  • [27] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
  • [28] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_forma-2014-0007
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