Topological foundations of an asymptotic approximation theory for sequences of matrices with increasing size

Sequences of matrices with increasing size arise in several contexts, including the discretization of integral and differential equations. An asymptotic approximation theory for this kind of sequences has recently been developed, with the aim of providing tools for computing their asymptotic singular value and eigenvalue distribution. The basis of this theory is the notion of approximating classes of sequences (a.c.s.), which is also fundamental to the theory of generalized locally Toeplitz sequences and hence to the spectral analysis of PDE discretization matrices. In this paper we show that the a.c.s. notion is a convergence notion induced by a pseudometrizable topology. We also identify a corresponding pseudometric and we study some of its properties. It turns out that there exists a strong connection between the a.c.s. topology and the topology of convergence in measure.