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  "body": "\n## Summary\n\nThe weak operator topology is a topology on the bounded operators on a Hilbert space.\nIn it, a net of operators converges when all its matrix coefficients $\\langle T x, y \\rangle$ converge.\nIt is much coarser than the norm topology, so more sets are compact and more nets converge.\nIt is the topology used to define [[von Neumann algebras|PSFWQVZ0FS]], and its closed unit ball is compact.\n\n## Explanations\n\n### L1 human\n\nAn operator on a Hilbert space can be tested by feeding it a vector $x$ and reading off one component $\\langle Tx, y \\rangle$ of the output.\nIn the weak operator topology, operators are close when these numbers are close for finitely many chosen $x$ and $y$.\nThis is a very forgiving notion of closeness.\nFor example, the right shift $S$ on sequences moves each entry one step to the right, so the powers $S^n$ push every vector away to infinity.\nEvery fixed component of $S^n x$ then tends to $0$, so $S^n \\to 0$ in this topology, although each $S^n$ preserves lengths.\n",
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