The Smolyak algorithm [\textit{S. A. Smolyak}, Dokl. Akad. Nauk SSSR 148, 1042--1045 (1963; Zbl 0202.39901)] finds approximants for a linear tensor product operator \(S=\bigotimes_{k=1}^d S_k: \bigotimes_{k=1}^d F_k \to \bigotimes_{k=1}^d G_k\) based on algorithms approximating the linear operators \(S_k: F_k\to G_k\) where \(F_k\) and \(G_k\) are functional Banach spaces. It was originally designed to approximate linear functionals of smooth multivariate functions like integrals. Here it is shown that the algorithm can also be applied to approximation problems on function classes like Sobolev spaces or even Hilbert spaces. For the applicability of the algorithms, it is essential that \(\| T_1\otimes~\cdots~\otimes T_d\| \leq\prod_{k=1}^d \| T_k\| \). It is proved in this paper that this norm inequality does hold for certain Banach spaces when appropriate norms and the ``natural'' tensor products are used. Applications in multivariate integration are given for which estimates for the approximation error are computed.