A Relation Between Pointwise Convergence of Functions and Convergence of Functionals

著者: Haïm Brézis, Élliott H. Lieb - Inequalities 1983 被引用: 2,050

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 ∫