# Definition of $C_0$

This is probably a silly question, but a couple of people that I have talked to have had different responses.

Does $C_0$ denote the set of continuous functions with compact support or the set of continuous functions which vanish at infinity?

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I have always seen $C_0(X)$ denoting the continuous functions vanishing at infinity, and $C_c(X)$ or $C_{00}(X)$ denoting the continuous functions with compact support, where $X$ is usually a locally compact Hausdorff space.

A special case is $c_0$, which is shorthand for $C_0(\mathbb{N})$, and $c_{00}$ means $C_{00}(\mathbb{N})$. In this case $\mathbb{N}$ has the discrete topology, and "continuous" is redundant.

By analogy, sometimes the compact operators on a Hilbert space are denoted by $B_0(H)$, and the finite rank operators by $B_{00}(H)$.

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The reason people have different responses is that the notation is not completely standardized. For example, Reed & Simon use $C_0^{\infty}(X)$ for smooth functions with compact support in a space $X$, and $C_{\infty}(X)$ for continuous functions vanishing at infinity. (But instead of $C_0(X)$ for continuous functions with compact support, they write $\kappa(X)$ for some reason...)

So you just have to check in every case which convention the text that you're reading uses.

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This is a good point. Takesaki also uses $C_\infty(X)$ for continuous functions vanishing at infinity in his operator algebras books. However, I have still never seen $C_0(X)$ used to denote continuous functions with compact support. – Jonas Meyer Oct 13 '10 at 7:36
@JonasMeyer : Both Duistermaat + Kolk, Distributions, and Blanchard + Brüning, Mathematical Methods in Physics, use $C_0(X)$ for the continuous functions with compact support. Hörmander's series The Analysis of Linear Partial Differential Operators comes close, using $C_0^k(X)$ for the $C^k$ functions with compact support. My observation is that these spaces of compactly supported functions, when denoted as $C_0$ with or without a superscript symbol, always seems have the inductive limit of the Fréchet topologies. – epimorphic Aug 12 '15 at 17:36