# nLab Lagrangian subspace

Contents

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Definition

In a symplectic vector space a Lagrangian subspace is a maximal isotropic subspace:

a sub-vector space

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 spacecondition 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)

Last revised on January 2, 2015 at 19:47:20. See the history of this page for a list of all contributions to it.