M(Lo), because, from M(Lo) = M'(Lo), it follows that M-IM' E GLo. In a similar vein, we can describe another family of interesting subspaces of a given symplectic space. Fix T (L), for a given Lagrangian subspace L C V, to be the collection of all Lagrangian subspaces L' transverse to L, so that T(L):= {L' EL(V), LPL'=V}.

Aw#0. Although these statements are left unproved, we will not hesitate to use them later in the text. In parallel to the situation in Euclidian geometry, we may form the following definition in symplectic geometry. 6. Two vectors v, w from the symplectic vector space (L; w) are called w-orthogonal, skew-orthogonal or - when there is no doubt about which w is intended - simply orthogonal, whenever w (v, w) = 0. This is also indicated by v 1 w. 1. 7. Ken with the symplectic form w given by w (v, v') _ (xi2 - xyi) for v = xlel + ...

### An Introduction to Symplectic Geometry (Graduate Studies in Mathematics, Volume 26) by Rolf Berndt

by Thomas

4.4