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 - \le... View More