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  "body": "\n## Summary\n\nThe numerical range of an operator $T$ on a complex Hilbert space is the set of values $\\ip{Tx}{x}$ for unit vectors $x$.\nIt is a convex subset of the complex plane whose closure contains the spectrum.\nIt gives easy bounds for the spectrum and for the norm, and it characterises self-adjoint operators as those with real numerical range.\n\n## Explanations\n\n### L1 human\n\nFor a unit vector $x$, the number $\\ip{Tx}{x}$ measures how much $Tx$ points back in the direction of $x$.\nIn quantum mechanics it is the expected value of the observable $T$ in the state $x$.\nCollecting these numbers over all unit vectors gives the numerical range.\nFor a real symmetric $2 \\times 2$ matrix with eigenvalues $1$ and $3$, the numerical range is the interval $[1, 3]$.\n",
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