Given a topological space \(X\). A function \(f\colon X\to \mathbb{R}\) vanishes at infinity if for every \(\varepsilon>0\) the set \(\{x\in X\mid \lvert f(x)\rvert>\varepsilon\}\) is compact .

The set of all functions vanishing at infinity is denoted by \(C_0(X)\).

See also Link to heading

• Fourier transform \(\mathcal{F}\colon L^1(\mathbb{R}^n)\to C_0(\mathbb{R}^n)\)