I have the usual definition of non degeneracy for a differential 2-form in a vector space V.
For every $v \in V:$ $\omega(v,u)=0 \,, \forall u \in V$ $\Rightarrow v=0$
and the definition given in my lecture:
$\forall x \neq 0\ \exists u \in V: \omega(v,u)=0$
the second definition immediately follows from the first one, but I am having trouble showing the other side, if this statement is even correct at all. Leading to my second question
How is the symplectic form non degenerate, if there exists a subspace $U \subset V$ such that,
$U^{\perp_{\omega}}=\{v \in V |\omega(v,u)=0\quad \forall u \in U\}$.
especially considering that $\omega$ is antisymmetric it would follow that
$\forall x \neq 0: \omega(x,x)=0$, which would mean we cannot conclude that $x$ has to be $0$.
Can anyone enlighten me where I am going wrong, since I am sure, the literature is correct this time.
Edit:
I already noticed my mistakes. Assume $\omega(v,u)=0$ for all $u \in V$, but since $\forall u \in V :\exists v \in V: \omega(v,u)\neq 0$, the assumption must be false.
The second question is along the same vein, that i have not considered, that it needs to hold for all elements in V.