Journal of Formalized Mathematics
Volume 1, 1989
University of Bialystok
Copyright (c) 1989
Association of Mizar Users
Construction of a bilinear symmetric form
in orthogonal vector space

Eugeniusz Kusak

Warsaw University, Bialystok

Wojciech Leonczuk

Warsaw University, Bialystok

Michal Muzalewski

Warsaw University, Bialystok
Summary.

In this text we present unpublished results by Eu\ge\niusz Ku\sak and
Wojciech Leo\'nczuk. They contain an axiomatic description of the class of all
spaces $\langle V$; $\perp_\xi \rangle$, where $V$ is a vector
space over a field F, $\xi: V \times V \to F$ is a bilinear symmetric form
i.e. $\xi(x,y) = \xi(y,x)$
and $x \perp_\xi y$ iff $\xi(x,y) = 0$
for $x$, $y \in V$.
They also contain an effective construction of bilinear symmetric
form $\xi$ for given orthogonal space $\langle V$; $\perp \rangle$
such that $\perp = \perp_\xi$. The
basic tool used in this method is the notion of orthogonal projection
J$(a,b,x)$ for $a,b,x \in V$.
We should stress the fact that axioms of orthogonal and symplectic
spaces differ only by one axiom, namely:
$x\perp y+\varepsilon z \>\&\> y\perp z+\varepsilon x \Rightarrow
z\perp x+\varepsilon y.$
For $\varepsilon=1$ we get the axiom on three perpendiculars characterizing
orthogonal geometry. For $\varepsilon=+1$ we get the axiom characterizing
symplectic geometry  see [5].
Supported by RPBP.III24.C6.
MML Identifier:
ORTSP_1
The terminology and notation used in this paper have been
introduced in the following articles
[6]
[3]
[8]
[1]
[2]
[7]
[4]
[9]
[5]
Contents (PDF format)
Bibliography
 [1]
Czeslaw Bylinski.
Functions and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
 [2]
Czeslaw Bylinski.
Functions from a set to a set.
Journal of Formalized Mathematics,
1, 1989.
 [3]
Czeslaw Bylinski.
Some basic properties of sets.
Journal of Formalized Mathematics,
1, 1989.
 [4]
Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski.
Abelian groups, fields and vector spaces.
Journal of Formalized Mathematics,
1, 1989.
 [5]
Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski.
Construction of a bilinear antisymmetric form in symplectic vector space.
Journal of Formalized Mathematics,
1, 1989.
 [6]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
 [7]
Wojciech A. Trybulec.
Vectors in real linear space.
Journal of Formalized Mathematics,
1, 1989.
 [8]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
 [9]
Edmund Woronowicz.
Relations defined on sets.
Journal of Formalized Mathematics,
1, 1989.
Received November 23, 1989
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