Sampling projections in the uniform norm

We show that there are sampling projections on arbitrary $n$-dimensional subspaces of $B(D)$ with at most $2n$ samples and norm of order $\sqrt{n}$, where $B(D)$ is the space of complex-valued bounded functions on a set $D$. This gives a more explicit form of the Kadets-Snobar theorem for the uniform norm and improves upon Auerbach’s lemma. We discuss consequences for optimal recovery in $L_p$.