  Given a Hilbert space $\mathcal H$ and a finite measure space $\Omega$, the
approximation of a vector-valued function $f: \Omega \to \mathcal H$ by a
$k$-dimensional subspace $\mathcal U \subset \mathcal H$ plays an important
role in dimension reduction techniques, such as reduced basis methods for
solving parameter-dependent partial differential equations. For functions in
the Lebesgue-Bochner space $L^2(\Omega;\mathcal H)$, the best possible subspace
approximation error $d_k^{(2)}$ is characterized by the singular values of $f$.
However, for practical reasons, $\mathcal U$ is often restricted to be spanned
by point samples of $f$. We show that this restriction only has a mild impact
on the attainable error; there always exist $k$ samples such that the resulting
error is not larger than $\sqrt{k+1} \cdot d_k^{(2)}$. Our work extends
existing results by Binev at al. (SIAM J. Math. Anal., 43(3):1457-1472, 2011)
on approximation in supremum norm and by Deshpande et al. (Theory Comput.,
2:225-247, 2006) on column subset selection for matrices.
