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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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@joriki Please consider converting your comment into an answer, so that this question gets removed from the unanswered tab. If you do so, it is helpful to post it to this chat room to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see here, here or here. –  Julian Kuelshammer May 5 at 9:20

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