Vanishing Dirichlet energy by dilation on the line
= Vanishing Dirichlet energy by dilation on the line
For $\phi\in H^1(\mathbb R)$ with $\lVert\phi\rVert_2=1$, the dilation
$$
u_R(x)=R^{-1/2}\phi(x/R)
$$
preserves the $L^2$ norm and satisfies $\lVert u_R'\rVert_2^2=R^{-2}\lVert\phi'\rVert_2^2$. Hence normalized functions on the line have Dirichlet-energy infimum zero, which no nonzero $L^2$ function attains.