I wanna know how to prove that a function involving a cartesian product of quotient space is well-defined.
Let's see this question: Bilinear form and quotient space
In that question, I guess that $g$ is well-defined if $(u_1+U_0,v_1+V_0) = (u_2+U_0, v_2+V_0)$ implies $g(u_1+U_0,v_1+V_0) = g(u_1+U_0,v_1+V_0)$, that is, $f(u_1,v_1) = f(u_2,v_2)$, whenever we have $u_1-u_2 \in U_0$ and $v_1-v_2 \in V_0$. But I don't know how to get there, can you help me??
Notice that if $u_1-u_2 \in U_0$ then $f(u_1-u_2,v)=0$ for any $v \in V$, that is, $f(u_1,v) = f(u_2,v)$ for any $v \in V$; and, if $v_1-v_2 \in V_0$ then $f(u,v_1-v_2) = 0$ for any $u \in U$, that is, $f(u,v_1) = f(u,v_2)$ for any $u \in U$. So $$f(u_1,v_1) = f(u_2,v_1) = f(u_2,v_2).$$