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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A relation between pointwise convergence of functions and convergence of functionals


Authors: Haïm Brézis and Elliott Lieb
Journal: Proc. Amer. Math. Soc. 88 (1983), 486-490
MSC: Primary 28A20; Secondary 46E30, 49A99
DOI: https://doi.org/10.1090/S0002-9939-1983-0699419-3
MathSciNet review: 699419
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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.


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Keywords: Convergence of functionals, pointwise convergence, <IMG WIDTH="29" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${L^p}$"> spaces
Article copyright: © Copyright 1983 American Mathematical Society