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  "body": "\n## Definition\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]] with inner product $\\ip{\\cdot}{\\cdot}$.\nLet $T$ be a [[densely defined|BCJHNHKQFG]] [[symmetric operator|H971WGMV6C]] on $H$ with domain $D(T)$.\nAssume that $T$ is bounded below: there is a real number $c$ with $\\ip{Tx}{x} \\ge c \\norm{x}^2$ for all $x \\in D(T)$.\nOn $D(T)$, put\n$$\\ip{x}{y}_T = \\ip{Tx}{y} + (1 - c)\\ip{x}{y}.$$\nThis is an inner product on $D(T)$, and its norm satisfies $\\norm{x}_T \\ge \\norm{x}$.\nLet $H_T$ be the completion of $D(T)$ for $\\norm{\\cdot}_T$.\nThe inclusion of $D(T)$ into $H$ extends to an injective map from $H_T$ into $H$, so $H_T$ is a subspace of $H$.\nThe *Friedrichs extension* of $T$ is the operator $T_F$ with domain\n$$D(T_F) = \\set{x \\in H_T : \\text{there is } z \\in H \\text{ with } \\ip{x}{y}_T = \\ip{z + (1 - c)x}{y} \\text{ for all } y \\in H_T},$$\ngiven by $T_F x = z$.\n\n## Remarks\n\nThe operator $T_F$ is a [[self-adjoint|0NEGN5CRSC]] extension of $T$, and $\\ip{T_F x}{x} \\ge c \\norm{x}^2$ for all $x \\in D(T_F)$.\nAn equivalent description is that $T_F$ is the restriction of the [[adjoint|86B7HF79PQ]] $T^*$ to $D(T^*) \\cap H_T$.\nThe space $H_T$ is the form domain of $T_F$, and the [[sesquilinear form|054YEQ0Y8M]] $(x, y) \\mapsto \\ip{Tx}{y}$ extends to it.\n\nExample: on $H = L^2(0, 1)$, let $T f = -f''$ on smooth functions with compact support in $(0, 1)$.\nIntegration by parts gives $\\ip{Tf}{f} = \\int_0^1 \\abs{f'}^2 \\, dx \\ge 0$, so $c = 0$ works.\nThe space $H_T$ is the Sobolev space $H^1_0(0, 1)$, and $T_F f = -f''$ on $D(T_F) = H^2(0, 1) \\cap H^1_0(0, 1)$, the Dirichlet Laplacian.\n\nNon-example: on $L^2(0, \\infty)$, let $S f = -i f'$ on smooth functions with compact support in $(0, \\infty)$.\nIt is symmetric, but $\\ip{Sf}{f}$ is not bounded below by any $c \\norm{f}^2$, so the construction does not apply.\nIn fact the [[deficiency indices|G4GMSNAX8X]] of $S$ are unequal, so $S$ has no self-adjoint extension at all.\n",
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