A relation between pointwise convergence of functions and convergence of functional

Haim Brezis, Elliott Lieb

Research output: Contribution to journalArticle

1255 Scopus citations

Abstract

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 <p <∞. This result is also generalized in Theorem 2 to some functionals other than the Lp norm, namely F|j(fn)— 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.

Original languageEnglish (US)
Pages (from-to)486-490
Number of pages5
JournalProceedings of the American Mathematical Society
Volume88
Issue number3
DOIs
StatePublished - Jul 1983

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Keywords

  • Convergence of functionals
  • Lp spaces
  • Pointwise convergence

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