Kernel methods have been successful in many areas, including scientific computing and machine learning. In this talk, we will discuss the convergence of two kernel algorithms, namely the singular value expansion (SVE) and the Cholesky algorithm with a pivoting strategy. They are infinite-dimensional analogues of two famous matrix decompositions, the SVD and the pivoted Cholesky decomposition. While these algorithms are effective in practice, even their existence had not been fully established. We prove two surprising results which were previously unanticipated : (1) for continuous kernels, the SVE may not even exist pointwise; (2) for Lipschitz continuous kernels the Cholesky algorithm with complete pivoting converges uniformly. This is joint work with Alex Townsend.