A relation between pointwise convergence of functions and convergence of functional

Haim Brezis; Elliott Lieb

doi:10.1090/S0002-9939-1983-0699419-3

A relation between pointwise convergence of functions and convergence of functional

Haim Brezis
, Elliott Lieb

Mathematics

Research output: Contribution to journal › Article › peer-review

2072 Scopus citations

Abstract

We show that if {F_n} is a sequence of uniformly L^p-bounded functions on a measure space, and if F_n →F pointwise a.e., then lim_n__∞{||F_n||^p_p-|| F_n-F||^p_p } = ||F_n||^p_p for all 0 <p <∞. This result is also generalized in Theorem 2 to some functionals other than the Lp norm, namely F|j(f_n)— j(f_n—f)— j (F_n) I →0 f°r suitable/: C →C and a suitable sequence {f_n}. A brief discussion is given of the usefulness of this result in variational problems.

Original language	English (US)
Pages (from-to)	486-490
Number of pages	5
Journal	Proceedings of the American Mathematical Society
Volume	88
Issue number	3
DOIs	https://doi.org/10.1090/S0002-9939-1983-0699419-3
State	Published - Jul 1983

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

Keywords

Convergence of functionals
Lp spaces
Pointwise convergence

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10.1090/S0002-9939-1983-0699419-3

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AU - Lieb, Elliott

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N2 - We show that if {Fn} is a sequence of uniformly Lp-bounded functions on a measure space, and if Fn →F pointwise a.e., then limn_∞{||Fn||pp-|| Fn-F||pp } = ||Fn||pp for all 0 n)— j(fn—f)— j (Fn) I →0 f°r suitable/: C →C and a suitable sequence {fn}. A brief discussion is given of the usefulness of this result in variational problems.

AB - We show that if {Fn} is a sequence of uniformly Lp-bounded functions on a measure space, and if Fn →F pointwise a.e., then limn_∞{||Fn||pp-|| Fn-F||pp } = ||Fn||pp for all 0 n)— j(fn—f)— j (Fn) I →0 f°r suitable/: C →C and a suitable sequence {fn}. A brief discussion is given of the usefulness of this result in variational problems.

KW - Convergence of functionals

KW - Lp spaces

KW - Pointwise convergence

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DO - 10.1090/S0002-9939-1983-0699419-3

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JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 3

ER -

A relation between pointwise convergence of functions and convergence of functional

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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