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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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    @joriki Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2015-05-05

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