Open Access

Daniel Kressner, Tingting Ni, André Uschmajew

• Given a Hilbert space H and a finite measure space Ω, the approximation of a vector-valued function f : Ω → H by a k-dimensional subspace U ⊂ 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 L2 (Ω; H), the best possible subspace approximation error d (2) k is characterized by the singular values of f. However, for practical reasons, 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 √ k + 1 · d (2) k . 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.