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  "body": "\n## Definition\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]] with inner product $\\ip{\\cdot}{\\cdot}$, and let $T \\colon H \\to H$ be a [[continuous linear map|03TM135G0N]].\nThe *numerical range* of $T$ is the set\n$$W(T) = \\set{\\ip{Tx}{x} : x \\in H, \\ \\norm{x} = 1} \\subseteq \\C.$$\nThe *numerical radius* of $T$ is $w(T) = \\sup \\set{\\abs{z} : z \\in W(T)}$.\n\n## Remarks\n\nThe numerical range is convex; this is the [[Toeplitz–Hausdorff theorem|8TV0DJBBAY]].\nThe closure of $W(T)$ contains the [[spectrum|BMKATJ802A]] of $T$.\nThe numerical radius satisfies $\\tfrac12 \\norm{T} \\le w(T) \\le \\norm{T}$, with the [[operator norm|06EKW1AKXF]].\nThe operator $T$ is self-adjoint if and only if $W(T) \\subseteq \\R$.\n\nExample: let $T = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ on $\\C^2$.\nFor $x = (x_1, x_2)$ with $\\abs{x_1}^2 + \\abs{x_2}^2 = 1$, $\\ip{Tx}{x} = x_2 \\overline{x_1}$, whose absolute value is at most $\\tfrac12$ and takes every value with $\\abs{z} \\le \\tfrac12$.\nSo $W(T)$ is the closed disc of radius $\\tfrac12$, and $w(T) = \\tfrac12 \\norm{T}$.\n\nNon-example: for the same $T$ and $x = (1, 1)$, $\\ip{Tx}{x} = 1$, which is not in $W(T)$; the vector $x$ is not a unit vector.\n",
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