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 \( H \).
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For a real or complex matrix, the three displayed conditions are equivalent to being a regular matrix summability method. For necessity, first apply the Uniform boundedness principle to each row's finite partial sums on the convergent sequence space, obtaining absolute summability of that row. Apply the principle again to the full row functionals, obtaining a uniform bound. Coordinate vectors and the constant-one sequence give the remaining conditions. For sufficiency, write and split the error into a finite head, controlled by columnwise convergence, and a uniformly small tail, controlled by the absolute row bound.