For the equation $Ax = b$ in the finite dimensional linear space one can apply Cramer's rule to find $x$ if operator $A$ is linear. If there is an equivalent or a similar method for an infinite dimensional spaces for a Cramer's rule or a determinant?
Say, $f,g\in \mathcal{C}[0,1]$ and $\mathcal{A}$ is a linear operator, then equation is $ \mathcal{A}f = g. $