A relation between pointwise convergence of functions and convergence of functionals

Abstract:

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