Central limit theorem: convergence in the norm $\|u\|=\bigl(\int_{-\infty}^\infty u^2(x)e^{\frac{x^2}2}\,dx\bigr)$

Sufficient conditions for the normal convergence in the topology specified in the title are given. These conditions hold for the uniformly and identically distributed variables, though there is no such a convergence for certain bounded and dentically distributed variables having regular densities.