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If I say, $f|_{k}$, what does that mean? Sorry to be short on words but I can not find it anywhere on google so maybe someone could explain it and what we typically use it for. I ran into the notation in a book and they do not explain what it is.

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  • $\begingroup$ In what book did you run into this notation? $\endgroup$ – awllower Oct 22 '14 at 0:47
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It means that if $f$ is defined on the set $X$, and $k \subseteq X$, then $f|_k$ is $f$ restricted to the set $k$.

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It might also have the meaning that $f$ is a modular form, and $k\in\operatorname{SL}_2(\mathbb Z),$ and $f\mid_k$ is $f$ under the action of $k.$
Compared to the answer of @Johanna, this shows that, if the context is missing, the meaning of this symbol is not fixed at all.
Hope this helps.

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