A Relation Between Pointwise Convergence of Functions and Convergence of Functionals
نویسندگان
چکیده
We show that if (J,,} is a sequence of uniformly LI-bounded functions on a measure space, and if.f, -fpointwise a.e., then lim,,_(I{lf,, 1 -IIf,, fII) If I,' for all 0 < p < oc. This result is also generalized in Theorem 2 to some functionals other than the L P norm, namely I. /( J,, -(f, f) f ) -1 0 for suitablej: C -C and a suitable sequence (fJ}. A brief discussion is given of the usefulness of this result in variational problems.
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تاریخ انتشار 2008