Vanishing Dirichlet energy by dilation on the line

For $\varphi \in H^1(\mathbb{R})$ with $\Vert \varphi {\Vert }_2=1$, the dilationpreserves the $L^2$ norm and satisfies $\Vert u_R^{\prime }{\Vert }_2^2=R^{-2}\Vert {\varphi }^{\prime }{\Vert }_2^2$. Hence normalized functions on the line have Dirichlet-energy infimum zero, which no nonzero $L^2$ function attains.