(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

3. 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 [itex] \dim Y = \frac12 \dim V [/itex] implies that [itex] \dim Y^\omega = \frac12 \dim V [/itex]. 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 [itex] Y \neq Y^\omega [/itex] then without loss of generality, we may assume there exists [itex] w \in Y \setminus Y^\omega [/itex] (otherwise, relabel [itex] Y [/itex] and [itex] Y^\omega [/itex]). The space [itex] W = \operatorname{span}(w) [/itex] is isotropic so [itex] W \subseteq W^\omega [/itex] and since [itex] W \subseteq Y [/itex] then [itex] Y^\omega \subseteq W^\omega [/itex]. I would like to put this together somehow, but I'm having trouble doing it.

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# Homework Help: Lagrangian subspaces of symplectic vector spaces

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