# Help with some notation in QFT

I'm reading a paper on QFT and QEIs but i'm a little sketchy on some of the prerequisites. Can anyone tell me what this represents, $$C_{0}^{\infty}(M)$$

Where M is globally hyperbolic spacetime. I understand it has something to do with its topology but im not sure what.

This is the space of all smooth functions on $M$ that vanish at $\infty$. Let me make this more precise.
As $M$ is a space-time manifold, there is an open cover $\mathcal{U}$ of $M$ and a $\mathcal{U}$-sequence of embeddings $\left( \phi_{U}: U \to \mathbb{R}^{4} \right)_{U \in \mathcal{U}}$ such that for all $U,V \in \mathcal{U}$, $$\phi_{U} \circ \phi_{V}^{-1}: \quad \mathbb{R}^{4} \supseteq {\phi_{V}}[U \cap V] \to {\phi_{U}}[U \cap V] \subseteq \mathbb{R}^{4}$$ is a smooth function between open subsets of $\mathbb{R}^{4}$.
Now, to say that $f \in {C_{0}}(M)$ means that
• $f: M \to \mathbb{R}$;
• $f \circ \phi_{U}^{-1}: \mathbb{R}^{4} \supseteq {\phi_{U}}[U] \to \mathbb{R}$ is a smooth function for each $U \in \mathcal{U}$; and
• for any $\epsilon > 0$, there exists a compact subset $K$ of $M$ such that $|f(x)| < \epsilon$ for all $x \in M \setminus K$.
• Note: Some authors use ${C_{0}^{\infty}}(M)$ to denote the space of compactly supported smooth functions on $M$. However, this is not very common. Instead, one usually writes ${C_{c}^{\infty}}(M)$ or ${C_{00}^{\infty}}(M)$. Nov 21, 2014 at 0:11
Generally speaking, $C^\infty_0(M)$ denotes the subset of $C^\infty(M)$ (smooth functions on $M$) consisting of compactly supported functions. [It is standard notation to use $C^\infty_0$ (as e.g. in the standard reference "Linear PDO's" by L. Hörmander), which is at least as common as $C^\infty_c$, whereas NO ONE uses $C^\infty_{00}$.]