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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 $$ \mathcal{A}f = g. $$

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There is a relevant discussion here: forums.xkcd.com/viewtopic.php?f=17&t=53455 –  joriki Apr 14 '11 at 11:38
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