@MISC{SYMSP_1.MIZ,
  AUTHOR = {Kusak, Eugeniusz and Leo\'nczuk, Wojciech and Muzalewski, Micha{\l}},
  TITLE = {Construction of a bilinear antisymmetric form
in symplectic vector space},
  NOTE = {Supported by RPBP.III-24.C6.},
  DAY = {23},
  MONTH = {November},
  YEAR = {1989},
  ADDRESS1 = {Warsaw University\\Bia{\l}ystok},
  ADDRESS2 = {Warsaw University\\Bia{\l}ystok},
  ADDRESS3 = {Warsaw University\\Bia{\l}ystok},
  SUMMARY = {  In this text we will present unpublished results by Eu\-ge\-niusz Ku\-sak.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 \cite{ORTSP_1.ABS}.}}


