A Relation Between Pointwise Convergence of Functions and Convergence of Functionals
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 for all 0 < p < ∞. This result is also generalized in Theorem 2 to some functional other than the L p norm, namely → 0 for suitable j: C → C and a suitable sequence f n. A brief discussion is given of the usefulness of this result in variational problems.
Key wordsConvergence of functional pointwise convergence Lp spaces
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