Suppose a linear vector space $\mathbb{V}$ that has a two-dimensional column matrix as vector. $$|u\rangle \rightarrow \ (x\ \ \ \ \ y)^T$$ Suppose the dual space of $\ \mathbb{V}$ have row vectors so that $$\langle u|\ \rightarrow \ \ (Ax+By\ \ \ \ Cx+Dy)$$ Now suppose a linear map $\mathcal{L}: \mathbb{V}\rightarrow \ \mathbb{R}$ such that $$(Ax+By\ \ \ Cx+Dy)\begin{pmatrix} x \\ y \end{pmatrix} =Ax^2+Dy^2+(B+C)xy$$ I want to find out the values or constraint on values of $A, B, C, D$ such that this linear map represent an inner product on $\mathbb{V}$.
So far, I have taken specific values of $x,y$ to show that $A,B$ should be positive. For example :
$$\langle u|u\rangle =A(1)^2+D(0)^2+(B+C)(1)(0)=A>0$$ Similarly, $B>0$. Further If you take, $x=y=1$. $$A+D+B+C>0$$ and $x=1=-y$ $$A+D-(B+C)>0\rightarrow -(A+D)<B+C<A+D$$ What I can do now? Any help is appreciable.