# Notation: What's meant by $C^{\infty}_{0}(\mathbb{R}^{+})$?

In Chapter 0 of Iwaniec's Spectral Methods of Automorphic Forms Iwaneic uses the notation $C^{\infty}_{0}(\mathbb{R}^{+})$ without definition.

I assume that it's the set of infinitely differentiable functions from $\mathbb{R} \to \mathbb{R}$ with range a subset of the positive reals and which tend toward zero "sufficiently quickly," but I don't know whether my guess is right or what the precise definition of "sufficiently quickly" is in this context.

What does the subscript of $0$ refer to?

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$C_b(\mathbb{R})$ (resp. $C_0(\mathbb{R})$) are the continuous functions from $R$ to $R$ that are bounded (resp. vanish at infinity).
This is one of the unfortunate cases where the notation can mean two different things. It can either be the compactly supported smooth functions on the positive reals, or it can be the smooth functions which tend to $0$ when $x \to 0$ or $x \to \infty$. I personally prefer to use a $c$ subscript for the former and reserve the $0$ subscript for the latter.