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?

Solution 1:

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 $B_{00}(H)$.

See this Springer Online Reference Works article.

Solution 2:

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.