We prove by induction on $n = \frac{1}{2}\dim V$ that every symplectic vector space $(V, \omega)$ admits a symplectic basis.

Base Case ($n = 1$, $\dim V = 2$):

Since $\omega$ is non-degenerate, there exist $v, w \in V$ with $\omega(v, w) \neq 0$. Rescaling, we may assume $\omega(v, w) = 1$. Set $e_1 = v$ and $f_1 = w$. Then:

$\omega(e_1, f_1) = 1, \quad \omega(e_1, e_1) = 0, \quad \omega(f_1, f_1) = 0$

So $\{e_1, f_1\}$ is a symplectic basis.

Inductive Step: Assume the theorem holds for symplectic spaces of dimension $2(n-1)$.

Let $(V, \omega)$ have dimension $2n$. Choose any non-zero $v_1 \in V$. By non-degeneracy, there exists $w \in V$ with $\omega(v_1, w) \neq 0$. Normalizing, choose $f_1$ so that $\omega(v_1, f_1) = 1$. Set $e_1 = v_1$.

Define $W = \text{span}\{e_1, f_1\}$. Then:

• $W$ is a symplectic subspace: $\omega|_W$ is non-degenerate
• $\dim W = 2$

Consider the symplectic complement:

$W^\omega = \{v \in V : \omega(v, w) = 0 \text{ for all } w \in W\}$

We claim that $\dim W^\omega = 2n - 2$ and $V = W \oplus W^\omega$.

Proof of claim:

1. Non-degeneracy of $\omega|_{W^\omega}$: Suppose $v \in W^\omega$ satisfies $\omega(v, u) = 0$ for all $u \in W^\omega$. Then $\omega(v, w) = 0$ for all $w \in W$ (by definition of $W^\omega$) and for all $u \in W^\omega$, so $\omega(v, x) = 0$ for all $x \in V$. By non-degeneracy, $v = 0$.

2. Direct sum decomposition: We have $W \cap W^\omega \subseteq W$ and $W \cap W^\omega \subseteq W^\omega$. If $v \in W \cap W^\omega$, then $v \in W$ and $\omega(v, w) = 0$ for all $w \in W$. Since $\omega|_W$ is non-degenerate, $v = 0$. Thus $W \cap W^\omega = \{0\}$.

3. Dimension count: By linear algebra, $\dim W^\omega = 2n - \dim W = 2n - 2$. Thus:

$\dim(W + W^\omega) = \dim W + \dim W^\omega = 2 + (2n-2) = 2n = \dim V$

Therefore $V = W \oplus W^\omega$.

By the inductive hypothesis, $(W^\omega, \omega|_{W^\omega})$ admits a symplectic basis $(e_2, \ldots, e_n, f_2, \ldots, f_n)$.

The union $(e_1, e_2, \ldots, e_n, f_1, f_2, \ldots, f_n)$ is a symplectic basis for $V$:

• $\omega(e_i, e_j) = 0$ for all $i, j$ (since each $e_i$ is in either $W$ or $W^\omega$, both isotropic)
• $\omega(f_i, f_j) = 0$ for all $i, j$ (similarly)
• $\omega(e_i, f_j) = \delta_{ij}$ (by construction for $i=1$ or $j=1$, and by induction for $i, j \geq 2$)

This completes the induction.