I was wondering whether I am correct that the second derivative of the dot product is this:
Let $f:X \times X \rightarrow \mathbb{K}$ $(x_1,x_2) \mapsto \langle x_1,x_2 \rangle$ Then we have $f'(x_1,x_2) (h_1,h_2)= f(x_1,h_2) + f(h_1,x_2)$ and consequently I woul say $f''(x_1,x_2)[(w_1,w_2),(w_3,w_4)]= f(w_3,w_2)+f(w_1,w_4)$. WHat makes me thinking that this is wrong is, that it does no longer depend on $(x_1,x_2)$.