A relation between pointwise convergence of functions and convergence of functionals

We show that if { f n } \left \{ {{f_n}} \right \} is a sequence of uniformly L p {L^p} -bounded functions on a measure space, and if f n → f {f_n} \to f pointwise a.e., then lim n → ∞ { ‖ f n ‖ p p − ‖ f n − f ‖ p p } = ‖ f ‖ p p {\lim _{n \to \infty }}\left \{ {\left \| {{f_n}} \right \|_p^p - \left \| {{f_n} - f} \right \|_p^p} \right \} = \left \| f \right \|_p^p for all 0 > p > ∞ 0 > p > \infty . This result is also generalized in Theorem 2 to some functionals other than the L p {L^p} norm, namely ∫ | j ( f n ) − j ( f n − f ) − j ( f ) | → 0 \int \left | {j({f_n}) - j({f_n} - f) - j(f)} \right | \to 0 for suitable j : C → C j:{\mathbf {C}} \to {\mathbf {C}} and a suitable sequence { f n } \left \{ {{f_n}} \right \} . A brief discussion is given of the usefulness of this result in variational problems.