Lagrangian subspaces of symplectic vector spaces

[Kreizhn]

Homework Statement

If (V,\omega) is a symplectic vector space and Y is a linear subspace with \dim Y = \frac12 \dim V show that Y is Lagrangian; that is, show that Y = Y^\omega where Y^\omega is the symplectic complement.

The Attempt at a Solution

This is driving me crazy since I don't think it should be that hard. If we try to go directly, the fact that \dim Y = \frac12 \dim V implies that \dim Y^\omega = \frac12 \dim V. At this point, it is actually sufficient to show that Y is either isotropic or co-isotropic, since dimensional arguments will give equality. However, I cannot see why this is the case.

On the other hand, if we assume that Y \neq Y^\omega then without loss of generality, we may assume there exists w \in Y \setminus Y^\omega (otherwise, relabel Y and Y^\omega). The space W = \operatorname{span}(w) is isotropic so W \subseteq W^\omega and since W \subseteq Y then Y^\omega \subseteq W^\omega. I would like to put this together somehow, but I'm having trouble doing it.

 

[Kreizhn]

Nevermind. I've concluded that the question as phrased was unsolvable. Indeed, consider \mathbb R^4 with the standard symplectic basis \{x_i,y_i\}. The subspace generated by x1 and y1 is two-dimensional, but is in fact symplectic rather than Lagrangian. Hence the statement cannot be true in general.