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Please let me know the meaning of the notation $ C^q_c(0,1)$

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    I would just cross out "with compact support in $(0,1)$" and probably I'd write $f:[0,1] \to \mathbb{R}$ since the requirement that $f$ be defined on all of $\mathbb{R}$ doesn't add anything to the content of the question.2011-04-17

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I recommend you to read Wikipedia article about Function Space.

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    Best suited as a comment.2018-01-17