### 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) |
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Pages (from-to) | 486-490 |

Number of pages | 5 |

Journal | Proceedings of the American Mathematical Society |

Volume | 88 |

Issue number | 3 |

DOIs | |

State | Published - Jul 1983 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

- Convergence of functionals
- Lp spaces
- Pointwise convergence

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## Cite this

Brezis, H., & Lieb, E. (1983). A relation between pointwise convergence of functions and convergence of functional.

*Proceedings of the American Mathematical Society*,*88*(3), 486-490. https://doi.org/10.1090/S0002-9939-1983-0699419-3