On the Brezis–Lieb lemma without pointwise convergence

Brezis–Lieb lemma is an improvement of Fatou Lemma that evaluates the gap between the integral of a functional sequence and the integral of its pointwise limit. The paper proves some analogs of Brezis–Lieb lemma without assumption of convergence almost everywhere. While weak convergence alone brings no conclusive estimates, a lower bound for the gap is found in L^p, p ≥ 3, under condition of weak convergence and weak convergence in terms of the duality mapping. We prove that the restriction on p is necessary and prove few related inequalities in connection to weak convergence