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  "body": "\n## Summary\n\nA differential equation in a [[Banach space|06GZ956657]] has an unknown $u(t)$ that is a vector of an infinite-dimensional space, such as a space of functions.\nThe basic example is the abstract Cauchy problem $u'(t) = A u(t)$, $u(0) = u_0$, with a linear operator $A$.\nMany partial differential equations of evolution type, such as the heat equation, take this form when the space variables are absorbed into the state space.\nThe linear theory rests on [[strongly continuous semigroups|EVF180QJXY]] of operators.\n\n## Explanations\n\n### L1 human\n\nThe heat in a rod is a whole temperature profile, a function of position.\nAs time passes, the profile changes, and its rate of change is given by its second derivative in space.\nSo one can view the heat equation as an ordinary differential equation for one moving point in a space of functions.\nWhen $A$ is a bounded operator, the solution is $u(t) = e^{tA} u_0$, given by the exponential series, just as for matrices.\nFor the heat equation $A$ is unbounded, and this is where the theory becomes deeper than for ordinary differential equations.\n",
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