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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 operator $T$ is *closed* if its [[graph|3RCAY97W1E]] $\\Gamma(T)$ is a [[closed subset|0ZCK8DQYN1]] of $X \\times Y$.\n\n## Remarks\n\nEquivalently, $T$ is closed when, for every sequence $(x_n)$ in $D(T)$ with $x_n \\to x$ in $X$ and $T x_n \\to y$ in $Y$, one has $x \\in D(T)$ and $Tx = y$.\n\nExample: on $X = Y = C[0, 1]$ with the supremum norm, let $D(T) = C^1[0, 1]$ and $Tf = f'$.\nIf $f_n \\to f$ and $f_n' \\to g$ uniformly, then $f_n(x) = f_n(0) + \\int_0^x f_n'$ gives $f(x) = f(0) + \\int_0^x g$, so $f \\in C^1[0, 1]$ and $f' = g$.\nHence $T$ is closed, although it is not bounded.\n\nNon-example: the same formula with $D(S)$ the space of polynomials is not closed.\nThe Taylor polynomials $p_n$ of $e^x$ converge uniformly to $e^x$ and $p_n' = p_{n-1}$ converges to $e^x$, but $e^x$ is not a polynomial, so it is not in $D(S)$.\n\nEvery bounded operator defined on all of $X$ is closed.\nIf $X$ and $Y$ are Banach spaces and $T$ is closed with $D(T) = X$, then $T$ is bounded, by the [[closed graph theorem|GXM8BX8YM7]].\n",
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