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  "body": "\n## Summary\n\nThe strong operator topology is the topology of pointwise convergence on the bounded operators on a Hilbert space.\nA net $T_i$ converges to $T$ when $T_i x \\to T x$ in norm for every vector $x$.\nIt lies between the [[weak operator topology|4XM1VN1C1N]] and the norm topology.\nIt is the natural topology for limits such as partial sums of orthogonal projections and for one-parameter unitary groups.\n\n## Explanations\n\n### L1 human\n\nTwo operators are close in the strong operator topology when they move each of finitely many chosen vectors to nearby places.\nThis is weaker than being close in norm, which asks for the same accuracy on all unit vectors at once.\nFor example, let $P_n$ keep the first $n$ coordinates of a sequence and set the rest to $0$.\nFor each fixed sequence $x$ in $\\ell^2$, the truncation $P_n x$ tends to $x$, so $P_n \\to I$ strongly.\nBut $P_n$ never comes close to $I$ in norm, since $I - P_n$ still keeps the $(n+1)$-st basis vector unchanged.\n",
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