 Silverman–Toeplitz theorem

The Silverman-Toeplitz theorem is a result in functional analysis and operator theory concerning the convergence of certain types of series of bounded linear operators. Specifically, it addresses the behavior of a series of projections in a Hilbert space. The theorem can be stated as follows: Let \( H \) be a Hilbert space and let \( \{ P_n \} \) be a sequence of orthogonal projections in \( H \).