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Jun 1, 2018 at 1:44 comment added Christian Remling We're of course dealing with Dirac's bra/ket notation here, and $|x\rangle\langle y|$ is the rank one operator $\langle y, \cdot\rangle x$. The series does converge to the identity strongly. The question is hardly suitable for MO though.
Jun 1, 2018 at 0:46 history answered Gerald Edgar CC BY-SA 4.0