= One-dimensional harmonic oscillator form domain
{title2=$X=H^1(\mathbb R)\cap L^2(\mathbb R,x^2\,dx)$}
This <Hilbert space> has norm $\|u\|_X^2=\|u\|_2^2+\|u'\|_2^2+\|xu\|_2^2$. Completeness follows from completeness of <H1 space> and closedness of the <multiplication operator> $u\mapsto xu$. Smooth cutoffs followed by convolution with a <mollifier> show that smooth compactly supported functions are dense. The space is the natural energy domain of the dimensionless <quantum harmonic oscillator>, whereas the operator domain additionally requires $u''$ and $x^2u$ to lie in <L2 space>.
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