  In this short note, we prove that the subspace of radial multipliers is
contractively complemented in the space of Fourier multipliers on the Bochner
space $\mathrm{L}^p(\mathbb{R}^n,X)$ where $X$ is a Banach space and where $1
\leq p <\infty$. Moreover, if $X = \mathbb{C}$, then this complementation
preserves the positivity of multipliers.
