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Recently I had to dive into abstract mathematics to understand deeply finite element method (I am an engineer not a mathematician). In some examples of linear spaces it appeared the space:

$C_{0}^{\infty}(\Omega)$ with $\Omega\subset$ in $\mathbb{R}^{d}$

The context says that this subspace of $\mathbb{R}^{d}$ is dense in $L^{p}(\Omega)$

Is this the set of continuous functions and derivatives that converges to '0'?

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  • $\begingroup$ $C^{\infty}$ is the set of infinitely differentiable functions $\endgroup$
    – Alex
    Commented Mar 4, 2015 at 11:34
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    $\begingroup$ I would guess this refers to the smooth functions that vanish at infinity, although it is always possible this particular author uses $0$ to denote some other property. $\endgroup$
    – AMPerrine
    Commented Mar 4, 2015 at 11:46
  • $\begingroup$ And where did that space appear? It's very probably defined somewhere in what you're reading... $\endgroup$ Commented Mar 5, 2015 at 8:57

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Depending on whom you ask, the notation $C_0^\infty$ means one of two things:

  1. The set of all infinitely differentiable functions $f$ such that $f=0$ outside of some compact set $K$
  2. The set of all infinitely differentiable functions $f$ such that $f$, and every derivative of $f$, tend to $0$ at infinity (or on the boundary of the domain, if we consider a domain instead of all $\mathbb{R}^n$).

People who subscribe to interpretation #2 use $C_c^\infty$ for the space from #1.

The space #2 is strictly larger than the space #1. One would have to see the book/paper to infer from the context which one is meant.

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  • $\begingroup$ It could also be the space of infinitely differentiable functions $f$ satisfying $f(0)=0$. $\endgroup$
    – user120513
    Commented Mar 4, 2015 at 14:25

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