In a symplectic vector space a Lagrangian subspace is a maximal isotropic subspace:
a sub-vector space
on which the restriction of the symplectic form vanishes;
and which has maximal dimension with this property.
Similarly for a symplectic manifold. See Lagrangian submanifold .
The collection of all Lagrangian subspaces of a given space is called its Lagrangian Grassmannian.
type of subspace $W$ of inner product space | condition on orthogonal space $W^\perp$ | |
---|---|---|
isotropic subspace | $W \subset W^\perp$ | |
coisotropic subspace | $W^\perp \subset W$ | |
Lagrangian subspace | $W = W^\perp$ | (for symplectic form) |
symplectic space | $W \cap W^\perp = \{0\}$ | (for symplectic form) |