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  "body": "\n## Definition\n\nLet $X$ and $Y$ be normed spaces, and let $T$ be an [[unbounded operator|KRJX5RA0PW]] from $X$ to $Y$ with domain $D(T)$.\nThe *graph* of $T$ is the set\n$$\\Gamma(T) = \\set{(x, Tx) : x \\in D(T)} \\subseteq X \\times Y.$$\nThe product $X \\times Y$ carries the norm $\\norm{(x, y)} = \\norm{x} + \\norm{y}$, or any equivalent norm.\n\n## Remarks\n\nIt is the [[graph|0JBJYT69ZV]] of the function $T \\colon D(T) \\to Y$.\nSince $T$ is linear, $\\Gamma(T)$ is a [[linear subspace|00YC1D652K]] of $X \\times Y$.\n\nExample: for $X = Y = \\R$ and $Tx = 3x$ with $D(T) = \\R$, the graph is the line $\\set{(x, 3x) : x \\in \\R}$.\n\nNon-example: the subspace $\\set{(0, y) : y \\in \\R}$ of $\\R \\times \\R$ is not the graph of any operator, since it contains $(0, 1)$, while every graph contains only $(0, 0)$ above $x = 0$.\n\nA linear subspace $G$ of $X \\times Y$ is the graph of an operator if and only if $(0, y) \\in G$ implies $y = 0$.\nThen $D(T)$ is the projection of $G$ onto $X$, and $Tx$ is the unique $y$ with $(x, y) \\in G$.\nIn this way $S \\subseteq T$ holds exactly when $\\Gamma(S) \\subseteq \\Gamma(T)$.\n",
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