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Proceedings of the American Mathematical Society
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
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\Vert {{f_n}} \right\Vert _p^p - \left\Vert {{f_n} - f} \right\Vert _p^p} \right\} = \left\Vert f \right\Vert _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\vert {j({f_n}) - j({f_n} - f) - j(f)} \right\vert \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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0699419-3
PII: S 0002-9939(1983)0699419-3
Keywords: Convergence of functionals, pointwise convergence, $ {L^p}$ spaces
Article copyright: © Copyright 1983 American Mathematical Society