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Jochen Wengenroth
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For every smooth function $g$ the linear partial differential equation with constant coefficients $P(D)f=g$ is solvable in convex sets. Although the statement has nothing to do with functional analysis Malgrange's proof heavily relied on Frechet space theory (and, of course, Fourier transformation).

The same holds if $g$ is a distribution (but one may object that distribution theory is part of functional analysis).