= Exponentially weighted L2 space
{title2=$L^2_\gamma=\{u:e^{\gamma x}u\in L^2\}$}
The exponentially weighted <Hilbert space> consists of measurable functions with $\int_{\mathbb R}|u(x)|^2e^{2\gamma x}\,dx<\infty$. Multiplication by $e^{\gamma x}$ is a unitary map onto <L2 space>. Requiring unweighted square integrability as well produces an intersection that is incomplete under the weighted norm alone. For example, $\mathbf1_{[-N,0]}$ is a weighted <Cauchy sequence> for $\gamma>0$, but its weighted limit $\mathbf1_{(-\infty,0]}$ is not in unweighted <L2 space>.
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