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  "body": "\n## Definition\n\nLet $H$ and $K$ be [[Hilbert spaces|HM2HHE3J51]] over $\\C$, with inner products $\\ip{\\cdot}{\\cdot}$ linear in the first argument.\nLet $T \\colon H \\to K$ be a [[continuous linear map|03TM135G0N]].\nThe *adjoint* of $T$ is the unique map $T^* \\colon K \\to H$ such that\n$$\\ip{Tx}{y}_K = \\ip{x}{T^* y}_H \\qquad \\text{for all } x \\in H \\text{ and } y \\in K.$$\n\n## Remarks\n\nFor fixed $y$, the map $x \\mapsto \\ip{Tx}{y}$ is a bounded linear functional on $H$, so the [[Riesz representation theorem|TTC40N49M6]] gives a unique vector $T^* y$ representing it.\nThe map $T^*$ is linear and bounded, with [[operator norm|06EKW1AKXF]] $\\norm{T^*} = \\norm{T}$ and $\\norm{T^* T} = \\norm{T}^2$.\nIt satisfies $(S + T)^* = S^* + T^*$, $(aT)^* = \\bar a T^*$, $(ST)^* = T^* S^*$ and $T^{**} = T$.\nFor finite-dimensional spaces it is the [[adjoint of a linear map|0R9NMSM1ZP]].\n\nExample: let $S(x_1, x_2, \\ldots) = (0, x_1, x_2, \\ldots)$ be the right shift on $\\ell^2$, and $L(y_1, y_2, \\ldots) = (y_2, y_3, \\ldots)$ the left shift.\nThen $\\ip{Sx}{y} = \\sum_{n \\ge 2} x_{n-1} \\overline{y_n} = \\sum_{m \\ge 1} x_m \\overline{y_{m+1}} = \\ip{x}{Ly}$, so $S^* = L$.\n\nNon-example: $S$ itself does not satisfy the defining identity in place of $S^*$.\nWith unit vectors $e_n$, $\\ip{S e_1}{e_2} = \\ip{e_2}{e_2} = 1$, while $\\ip{e_1}{S e_2} = \\ip{e_1}{e_3} = 0$.\nSo $S^* \\ne S$, and $S$ is not self-adjoint.\n\nOver $\\R$, the same definition applies, and then $(aT)^* = a T^*$.\n",
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