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  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|06GZ956657]] over $\\R$ or $\\C$ with norm $\\norm{\\cdot}$, and let $J \\subseteq \\R$ be an interval.\nA function $u \\colon J \\to X$ is *differentiable* at $t \\in J$ if the limit $u'(t) = \\lim_{h \\to 0} \\frac{u(t + h) - u(t)}{h}$ exists in the norm of $X$.\nLet $D \\subseteq X$ and $F \\colon J \\times D \\to X$.\nA *differential equation in $X$* is the equation $u'(t) = F(t, u(t))$.\nA *(classical) solution* on $J$ is a continuously differentiable $u \\colon J \\to X$ with $u(t) \\in D$ and $u'(t) = F(t, u(t))$ for all $t \\in J$.\nIn the main linear case, $F(t, u) = A u + f(t)$, where $A \\colon D(A) \\to X$ is a linear operator, $D = D(A)$ and $f \\colon J \\to X$.\nThe [[initial value problem|0FXDW992DS]] $u' = A u$, $u(0) = u_0$ is the *abstract Cauchy problem*.\n\n## Remarks\n\nExample: if $A$ is a bounded operator on $X$, then $u(t) = e^{tA} u_0 = \\sum_{k \\ge 0} \\frac{t^k}{k!} A^k u_0$ solves the abstract Cauchy problem on $\\R$.\nThe series converges in [[operator norm|06EKW1AKXF]] since $\\norm{A^k} \\le \\norm{A}^k$, and term-by-term differentiation gives $u' = A u$.\nFor $X = \\R^n$ this is the [[matrix exponential|020M04FN1H]] solution of a linear system of [[ordinary differential equations|0NT64JV96A]].\n\nNon-example: take $X = C([0, 1])$ with the supremum norm and $u(t) = (s \\mapsto \\abs{s - t})$ for $t \\in [0, 1]$.\nFor each fixed $s$, the number $u(t)(s)$ has a derivative in $t$ for $t \\neq s$, equal to $-\\sgn(s - t)$.\nBut the difference quotients $\\frac{u(t + h) - u(t)}{h}$ do not converge in the supremum norm: their pointwise limit $s \\mapsto -\\sgn(s - t)$ is not continuous for $0 \u003c t \u003c 1$.\nSo $u$ is not differentiable as a map into $X$, and it is not a classical solution of any differential equation in $X$.\n\nWhen $A$ is closed but unbounded, the Cauchy problem is well posed exactly when $A$ generates a strongly continuous semigroup $T(t)$, and then $u(t) = T(t) u_0$.\n\n## Explanations\n\n### L1 human\n\nThe unknown is a moving point in a Banach space, and its velocity is computed in the norm of that space.\nBounded operators behave like matrices; unbounded ones, such as derivatives in space, need more care.\n",
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