Playing with math sym. – David Davalos, PhD

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Let $\Delta$ an operator over the Hilbert space $\mathcal{H}$ , we call it bounded if and only if $||\Delta |\psi \rangle||\leq t|||\psi \rangle|| \ \ \forall t \geq 0 \ \ \forall |\psi \rangle \in \mathcal{H}$ , with $|||\psi\rangle||=\sqrt{\langle \psi | \psi \rangle}$ . The set of all bounded operators over $\mathcal{H}$ is denoted as $\mathcal{B}(\mathcal{H})$ . In particular the minimum $t$ for which the inequality above holds, is called the operator norm of $\Delta$ . This norm induces the operator norm topology. Such norm is typically too strong for many purposes, a very wide used one is the so called weak operator topology, it will be discussed later.