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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[normed spaces|03J5RAKB30]] over $\\R$ or $\\C$.\nAn *unbounded operator* (or *linear operator*) $T$ from $X$ to $Y$ is a pair consisting of\n1. a [[linear subspace|00YC1D652K]] $D(T)$ of $X$, the *domain* of $T$, and\n2. a [[linear map|0NRYHJPQG2]] $T \\colon D(T) \\to Y$.\n\nTwo such operators $S$ and $T$ are equal when $D(S) = D(T)$ and $Sx = Tx$ for all $x \\in D(T)$.\nThe operator $T$ is an *extension* of $S$, written $S \\subseteq T$, when $D(S) \\subseteq D(T)$ and $Tx = Sx$ for all $x \\in D(S)$.\n\n## Remarks\n\nExample: on $X = Y = L^2(0, 1)$, let $D(T)$ be the space of continuously differentiable functions on $[0, 1]$, and let $Tf = f'$.\nThis is linear, so $T$ is an unbounded operator.\nIt is not bounded: $f_n(x) = \\sin(n \\pi x)$ has $\\norm{f_n} = 1/\\sqrt{2}$, while $\\norm{T f_n} = n \\pi / \\sqrt{2}$.\n\nNon-example: the rule $f \\mapsto f'$ is not an operator with domain all of $L^2(0, 1)$, since many elements of $L^2(0, 1)$ have no derivative in $L^2(0, 1)$.\n\nEvery [[continuous linear map|03TM135G0N]] $X \\to Y$ is an unbounded operator with domain $X$; the word \"unbounded\" means \"not necessarily bounded\".\nSums and products are defined on the natural domains: $D(S + T) = D(S) \\cap D(T)$ and $D(ST) = \\set{x \\in D(T) : Tx \\in D(S)}$.\n",
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