I received this problem about Morse Index one day ago:
X is a linear space over C, q is a Hermitian form over X, V is a subspace of X. If q is nondegenerate and the restriction of q on V is 0, then the dimension of the maximal negative definite subspace of q is equal to the dimension of V.
I tried to prove the dimension of X is finite first, but I failed, which made me confused. Now I have no idea about how to prove it anymore, so I choose to ask for help here. If you have spare time, please help me. Thank you so much.
Denote the maximal negative (positive) definite subspace of q by $X^-$ ($X^+$). You can prove that $X=X^+ \oplus X^-$, since q is nondegenerate. Then you can try to prove that if assume $X=V \oplus U$, than $\dim V =\dim U$. (Construct a map.) Finally you will find that $m^-q=m^+q =\dim V$.
Hope can help you.