 Exponentially weighted L2 space

The exponentially weighted Hilbert space consists of measurable functions with ${\int }_{\mathbb{R}}|u(x){|}^2{e}^{2\gamma x}dx<\infty$. 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, $1_{[-N,0]}$ is a weighted Cauchy sequence for $\gamma >0$, but its weighted limit $1_{(-\infty ,0]}$ is not in unweighted L2 space.