Journal of Formalized Mathematics
Volume 1, 1989
University of Bialystok
Copyright (c) 1989 Association of Mizar Users

## Construction of a bilinear antisymmetric form in symplectic vector space

Eugeniusz Kusak
Warsaw University, Bialystok
Wojciech Leonczuk
Warsaw University, Bialystok
Michal Muzalewski
Warsaw University, Bialystok

### Summary.

In this text we will present unpublished results by Eu\-ge\-niusz Ku\-sak. It contains 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 antisymmetric form i.e. $\xi(x,y) = -\xi(y,x)$ and $x \perp_\xi y$ iff $\xi(x,y) = 0$ for $x$, $y \in V$. It also contains an effective construction of bilinear antisymmetric form $\xi$ for given symplectic 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 characterizing symplectic geometry. For $\varepsilon=-1$ we get the axiom on three perpendiculars characterizing orthogonal geometry - see [5].

Supported by RPBP.III-24.C6.

#### MML Identifier: SYMSP_1

The terminology and notation used in this paper have been introduced in the following articles [6] [3] [8] [1] [2] [7] [4] [9]

Contents (PDF format)

#### Bibliography

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