This notation is very useful when applying the Hilbert spaces in Quantum Theory. It exploits some properties of duality, eigenvalues/eigenvectors, projectors and self-adjoint operators. In mathematics, perhaps it is difficult to adopt, because mathematicians are using notations that are more general, and cannot exploit these particularities. But if you know Hilbert spaces you can learn this notation in one minute, and then it makes visible many of these nice properties.
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