Journal of Formalized Mathematics
Volume 9, 1997
University of Bialystok
Copyright (c) 1997 Association of Mizar Users

The Steinitz Theorem and the Dimension of a Real Linear Space


Jing-Chao Chen
Shanghai Jiaotong University, Shanghai

Summary.

Finite-dimensional real linear spaces are defined. The dimension of such spaces is the cardinality of a basis. Obviously, each two basis have the same cardinality. We prove the Steinitz theorem and the Exchange Lemma. We also investigate some fundamental facts involving the dimension of real linear spaces.

MML Identifier: RLVECT_5

The terminology and notation used in this paper have been introduced in the following articles [9] [8] [16] [10] [7] [2] [17] [4] [5] [1] [6] [3] [13] [15] [12] [11] [14]

Contents (PDF format)

  1. Prelimiaries
  2. The Steinitz Theorem
  3. Finite Dimensional Vector Spaces
  4. The Dimension of a Vector Space

Bibliography

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[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[6] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[7] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[8] Andrzej Trybulec. Enumerated sets. Journal of Formalized Mathematics, 1, 1989.
[9] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[10] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[11] Wojciech A. Trybulec. Operations on subspaces in real linear space. Journal of Formalized Mathematics, 1, 1989.
[12] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Journal of Formalized Mathematics, 1, 1989.
[13] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.
[14] Wojciech A. Trybulec. Basis of real linear space. Journal of Formalized Mathematics, 2, 1990.
[15] Wojciech A. Trybulec. Linear combinations in real linear space. Journal of Formalized Mathematics, 2, 1990.
[16] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[17] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received July 1, 1997

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