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Let $(V,\omega)$ be a symplectic vector space. Let $F \subseteq V$ be a subspace.

Show that $V$ admits a symplectic basis $\{e_1,\ldots,e_n,f_1,\ldots,f_n\}$ with the following properties:

(1) If $F$ is symplectic then $F=span\{e_1,\ldots,e_k,f_1,\ldots,f_k\}$ for some $k$

(2) If $F$ is isotropic then $F=span\{e_1,\ldots,e_k\}$ for some $k$

(3) If $F$ is co-isotropic then $F=span\{e_1,\ldots,e_n,f_1,\ldots,f_k\}$ for some $k$

(4) If $F$ is Lagrangian then $F=span\{e_1,\ldots,e_n\}$

The converse of these statements are easy to prove by applying definitions. But I have no idea how to prove these.

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I'm guessing this question is unanswered because it's an exercise and you haven't sketched any work on the problem. The ideas for all 4 are basically the same, and all ideas you need to do it is the linear algebra "tinkering" that you do in the (symplectic) Gram-Schmidt process. I would recommend reading a proof of the symplectic Gram-Schmidt theorem, then trying to prove these four statements using the same ideas. You can probably find a lot of places to read about symplectic GS, or I just wrote an explanation in my answer to $SP_{2n}(\mathbb {R})$ acts transitively on $\mathbb {R}^{2n}$. If you get stuck, you can update the question with your progress and we'll try to help you out.

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