Let $H$ be a Hilbert space and let $\{U_k\}_{k=1}^\infty$ be a sequence of closed subspaces of $H$. Are there any sufficient or neccessary criteria for the sum $\sum_{k=1}^\infty P_{U_k}$ to converge (in norm or pointwise)? If the subspaces $\{U_k\}_{k=1}^\infty$ are orthogonal to each other, then it is clear, but are there any other criteria (maybe $\omega$-linear indepencendy)?
2026-03-27 20:20:03.1774642803
When does a sum of orthogonal projectors converge?
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