The theory states that $$f(x,y)\\ \text{with} \ x\ \text{and}\ y \in R^n$$ in order to be a valid kernel, beside being symmetric, has to be an inner product in a suitable space.
Is this latter requirement equivalent to claim that the following Matrix:
$$ \begin{bmatrix} f(x,x) & f(x,y) \\ f(y,x) & f(y,y) \end{bmatrix} $$
is semidefinite positive for any pair x and y? If not, why?