The sequential closedness of \(\sigma\)-complete Boolean algebras of projections (Q1359580)

Let \(X\) be a locally convex Hausdorff space and let \(L(X)\) denote the space of all continuous linear operators of \(X\) into itself. If \(X\) is a separable Fréchet space, then any \(\sigma\)-complete (in the sense of Bade) Boolean algebra of projections is a strong operator closed subspace of \(L(X)\). It is no longer the case if \(X\) is non-separable, but then we can often replace closedness with sequential closedness. It is shown that under mild completeness assumptions on \(X\), one obtains the sequential closedness of a \(\sigma\)-complete atomic Boolean algebra of projections. The assumptions are satisfied if \(X\) is a Fréchet space. A counterexample is given for a non-atomic algebra.