= Range of the Volterra integration operator
{title2=$\mathcal R(K)=\{f\in H^1(0,1):f(0)=0\}$}
For $Ku(y)=\int_0^y u(x)\,dx$ on the <Hilbert space> $L^2(0,1)$, every image has a <weak derivative> $f'=u$ and zero trace at zero. Conversely every such <Sobolev space> element is this integral of its <weak derivative>. The range is dense because it contains smooth functions supported in $(0,1)$, but it is not closed: a step function can be approximated in $L^2$ by continuous ramps and is not itself in $H^1$. The <Moore–Penrose inverse of an operator> has this range as its domain and differentiates there.
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