# Lagrangian complement in symplectic vector space

Let $(V,\omega)$ be a vector space over $K$ together with symplectic form $\omega : V \times V \rightarrow K$. Let $U \subseteq V$ be a Lagrangian subspace (in other words, $U = U^{\perp}$). I want to know whether there must exist another Lagrangian subspace $W$ such that $U \oplus W = V.$

So far I have only been able to guarantee that an isotropic space $W$ exists such that $U \oplus W = V$; in other words, $W \subseteq W^{\perp}.$ I used Zorn's lemma to construct $W$.

If $V$ is finite-dimensional, then $\mathrm{dim} W = \mathrm{dim}(V) / 2 = \mathrm{dim}(W^{\perp})$ immediately implies that $W = W^{\perp}$.

Does anyone know what happens when $V$ is infinite-dimensional?

• Wow, out for a week and nothing! – user253055 Jul 12 '15 at 22:03