Given a topological space $X$ and a real (or complex) valued function $f$ on $X$, we say that $f$ vanishes at infinity if for any $\varepsilon>0$ there is a compact $K \subseteq X$ such that $|f(x)|<\varepsilon$ whenever $x \in X \setminus K$.
The set of all real (or complex) valued functions on $X$ that vanish at infinity with the norm defined by $\|f\|=sup\{|f(x)|:x \in X\}$ is a real (or complex) Banach space.
Is there today a standard notation for this space?
Rudin calls it $C_0(X)$ in his functional analysis book. Gillman calls it $C_\infty(X)$ in "Rings of continuous functions". Hewitt and Stromberg also call it $C_0(X)$ in theirs "Real and abstract analysis".