> Moreover any vector z can be uniquely expresseed as p+q where p is in W and v is in symp(W) Of course, crucially, this is only true if `W` is non-degenerate;

> Moreover any vector z can be uniquely expresseed as p+q where p is in W and v is in symp(W) Of course, crucially, this is only true if `W` is non-degenerate; as you'll know, one of the important things about symplectic spaces is the existence of half-dimensional, totally isotropic subspaces. (In fact, there's nothing particularly symplectic per se about this issue; it is not the (conjugate-)symmetry of the usual inner product so much as its positive definiteness that means that no such extra non-degeneracy condition must be imposed. Indeed, the study of non-positive definite but symmetric inner products, as in the geometry of relativistic spacetime, carries its own challenges to intuition.)