  Let $X(\mathbb{R})$ be a separable Banach function space such that the
Hardy-Littlewood maximal operator $M$ is bounded on $X(\mathbb{R})$ and on its
associate space $X'(\mathbb{R})$. Suppose $a$ is a Fourier multiplier on the
space $X(\mathbb{R})$. We show that the Fourier convolution operator $W^0(a)$
with symbol $a$ is compact on the space $X(\mathbb{R})$ if and only if $a=0$.
This result implies that nontrivial Fourier convolution operators on Lebesgue
spaces with Muckenhoupt weights are never compact.
