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  "body": "\n## Definition\n\nLet $X$ and $Y$ be normed spaces, and let $T$ be a [[closable operator|3TWMBFQJJ8]] from $X$ to $Y$.\nThe *closure* of $T$ is the operator $\\overline{T}$ whose [[graph|3RCAY97W1E]] is the [[closure|0QEAJB91GZ]] of the graph of $T$:\n$$\\Gamma(\\overline{T}) = \\overline{\\Gamma(T)} \\subseteq X \\times Y.$$\n\n## Remarks\n\nExplicitly, $x \\in D(\\overline{T})$ if and only if there is a sequence $(x_n)$ in $D(T)$ with $x_n \\to x$ such that $(T x_n)$ converges, and then $\\overline{T} x = \\lim T x_n$.\nThe operator $\\overline{T}$ is [[closed|VM7B38PA87]], it extends $T$, and it is contained in every closed extension of $T$.\n\nExample: on $C[0, 1]$ with the supremum norm, let $T$ be differentiation on polynomials.\nEvery $f \\in C^1[0, 1]$ is a limit of polynomials $p_n$ with $p_n' \\to f'$ uniformly: take polynomials $q_n \\to f'$ uniformly and set $p_n(x) = f(0) + \\int_0^x q_n$.\nCombined with the closedness of differentiation on $C^1[0, 1]$, this gives $D(\\overline{T}) = C^1[0, 1]$ and $\\overline{T} f = f'$.\n\nNon-example: an operator that is not closable has no closure, because the closure of its graph contains a pair $(0, y)$ with $y \\ne 0$ and is not a graph.\n\nA closed operator is its own closure.\nA subspace $D \\subseteq D(\\overline{T})$ with $\\overline{T|_D} = \\overline{T}$ is called a *core* of $\\overline{T}$.\n",
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