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  "body": "\n## Summary\n\nThe exponential of a bounded operator $T$ on a Banach space is the operator $e^T = \\sum_{n \\ge 0} T^n / n!$.\nIt extends the [[matrix exponential|020M04FN1H]] to infinite dimensions.\nThe family $t \\mapsto e^{tT}$ solves the linear differential equation $x' = Tx$ in the Banach space and is the simplest example of an operator semigroup.\n\n## Explanations\n\n### L1 human\n\nThe exponential of a number can be computed by the series $1 + t + t^2/2 + t^3/6 + \\cdots$.\nThe same series makes sense when $t$ is replaced by a bounded operator, because the factorials make the terms shrink fast.\nIf $T$ multiplies the $n$-th coordinate of a sequence by $\\lambda_n$, with $(\\lambda_n)$ bounded, then $e^T$ multiplies it by $e^{\\lambda_n}$.\nThis turns a rule of instantaneous change, $x' = Tx$, into the state at every later time, $x(t) = e^{tT} x(0)$.\n",
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