Necessary and sufficient conditions to have an inner product in $\mathbb R^2$ I'm trying to solve this question:

Given real numbers $a, b, c$, in order to exist  an inner product in
   $\mathbb R^2$ such that $\langle e_1,e_1\rangle=a$, $\langle
 e_1,e_2\rangle=\langle e_2,e_1\rangle=b$ and $\langle
 e_2,e_2\rangle=c$ is necessary and sufficient: $a \gt 0$ and $ac \gt
 b^2$.

I'm having troubles to prove the sufficient condition. If we have $a \gt 0$ and $ac \gt
 b^2$, we can add the other properties of the inner product (positiveness, linearity) in the element basis and finish the proof? is it simple as that?
Thanks in advance
 A: One can certainly proceed directly as you suggest. If you know the relevant facts, another option is this, which is a bit longer but which motivates the inequalities:
Given a basis $(f_a)$ of your vector space ($\mathbb{R}^2$ in your case) we can represent any bilinear form $\langle\,\cdot\, , \,\cdot\,\rangle$ as a matrix $A$ whose $(a, b)$ entry is $f_b^T A f_a = \langle f_a, f_b\rangle$. If the bilinear form is symmetric, then it is an inner product if $\langle x, x > 0$ for all nonzero vectors $x$. For any eigenvector $x_{\lambda}$ of $\lambda$, we have
$$\langle x, x \rangle = x^T A x = x^T (\lambda x) = \lambda x^T x,$$
so the bilinear form is an inner product if it is symmetric and if the eigenvalues of its matrix $A$ are all positive.
In our case, our basis is $(e_a)$ and the corresponding matrix is
$$A := \left(\begin{array}{cc}a & b\\b & c\end{array}\right).$$ The eigenvalues of $A$ are the roots of the characteristic polynomial
$$p_A(t) = \det(t I - A) = (t - a)(t - c) - b^2 = t^2 - (a + c) t + (ac - b^2).$$
The discriminant is that is if $$\Delta := (-(a + c))^2 - 4(1)(ac - b^2) = (a^2 + 2ac + c^2) - 4(ac - b^2) = (a - c)^2 + 4b^2,$$ which is always nonnegative, so both roots of $p_A$ are real. Both roots are positive iff $a + c$ is positive and $ac - b^2$ are both positive. If the latter holds and $a$ is positive, then the former holds, so a sufficient and necessary condition is that $a > 0$ and $ac > b^2$.
