= Sequential properness for a self-adjoint operator
For a bounded <self-adjoint operator> $L$, every bounded sequence whose images converge has a norm-convergent subsequence exactly when its kernel is finite-dimensional and its range is closed, or equivalently $0\notin\Sigma_{\mathrm e}(L)$ for the <essential spectrum of a bounded self-adjoint operator>. Split the sequence into kernel and kernel complement: finite dimensionality gives a subsequence on the first part and the <closed-range bound on the kernel complement> makes the second part a <Cauchy sequence>. Infinite kernel or approximate null unit vectors in its complement obstruct the property.
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