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.

翻译：给定一个希尔伯特空间 $\mathcal H$ 和一个有限测度空间 $\Omega$，利用 $k$ 维子空间 $\mathcal U \subset \mathcal H$ 逼近向量值函数 $f: \Omega \to \mathcal H$ 在降维技术中具有重要作用，例如求解参数相关偏微分方程的约化基方法。对于 Lebesgue-Bochner 空间 $L^2(\Omega;\mathcal H)$ 中的函数，最优子空间逼近误差 $d_k^{(2)}$ 由 $f$ 的奇异值刻画。然而，出于实际考虑，$\mathcal U$ 通常被限制为由 $f$ 的采样点张成。我们证明这一限制对可达到的误差影响较小：总存在 $k$ 个样本，使得所得误差不大于 $\sqrt{k+1} \cdot d_k^{(2)}$。我们的工作扩展了 Binev 等人 (SIAM J. Math. Anal., 43(3):1457--1472, 2011) 关于上确界范数逼近的现有结果，以及 Deshpande 等人 (Theory Comput., 2:225--247, 2006) 关于矩阵列子集选取的研究。