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My impression is that alot of notions in functional analysis are related to ordinary differential equations.

Are there situations or problems in the theory of differential equations that are in some sense "the example" or have some historical connection to the "fundamental theorems" of functional analysis.

The theorems I have in mind is Open Mapping, closed graph, Hahn-Banach, princical of uniform boundedness. Connections or ideas from "state/phase space" models would be nice.

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One such example would be the heat equation $\dot u - \Delta u=0$. The study of this equation in the context of functional analysis needs for instance

  1. The Laplace operator as unbounded operator on $L^2$. There, the closed graph theorem might be important.
  2. The study of the semigroup $e^{-t\Delta}$ generated by the Laplace. There, spectral theory could be of importance.
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Ascoli-Arzelà (not in the list, but can be reasonably considered an important theorem of functional analysis) can be used to prove the Cauchy–Peano theorem.

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