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

JingChao Chen

Shanghai Jiaotong University,
Shanghai
Summary.

Finitedimensional 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.
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]

Prelimiaries

The Steinitz Theorem

Finite Dimensional Vector Spaces

The Dimension of a Vector Space
Bibliography
 [1]
Grzegorz Bancerek.
Cardinal numbers.
Journal of Formalized Mathematics,
1, 1989.
 [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
[
Download a postscript version,
MML identifier index,
Mizar home page]