= Gaussian Sobolev space
{c}
{title2=$H^1(\gamma)=\{f:f,f'\in L^2(\gamma)\}$}
= Gaussian Sobolev form domain
{c}
{synonym}
The first Gaussian Sobolev space consists of functions and their weak first <derivatives> square-integrable for standard <Gaussian measure>. If $f=\sum c_nh_n$ in the normalized <Probabilists' Hermite polynomial> basis, <Gaussian integration by parts> gives $\langle f',h_{n-1}\rangle=\sqrt n c_n$. Thus <Parseval identity> identifies it with $\sum(1+n)|c_n|^2<\infty$, and polynomial truncations converge in its norm. It is the form domain of the <Gaussian number operator>, with <Gaussian Dirichlet energy> $\sum n|c_n|^2$.
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