Im choking with this exercise because of the indefinite scalar product. I know the process for the definite one.
The first thing I'm asked to do is to check GS is still valid for indefinite scalar product. I have to check this:
a) Check that if the subspace $W_k$ is non degenerate for all $k = 1, . . . , m$ then the GramSchmidt procedure works
What I have tried is to use the definition of a non degenerate subspace: Be $x \in W_k$ and $g(x,y)=0$ for all $y\in W_k \rightarrow x=0$
But I dont know how to follow. I have constructed $x_1$ and $x_2$ with the GS process and My intuition says that the definition of non-degenerate implies that the def. of the projectors are well done. Problem comes with a hint.
Hint. Let $S = (v_1, · · · , v_m) ⊂ V$ be a linearly independent set which spans a non-degenerate subspace $W = L(S)$ and put $W_k := L(v_1, . . . , v_k)$ for every $k = 1, . . . , m$