Vol. 38, No. 1, 1971

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Fatou’s lemma in normed linear spaces

Stephen Scheinberg

Vol. 38 (1971), No. 1, 233–238

This note presents a generalization of Fatou’s lemma to arbitrary normed linear spaces. Several examples illustrate the situations in which this notion is meaningful. The main theorem gives an abstract characterization of the Fatou property. In particular this resolves the case of any reflexive space. An example shows that Fatou’s lemma may fail even for uniform convergence in a normed algebra of continuous functions.

Frequently in analysis one obtains a function by a limiting process which is weaker (less demanding) than convergence in the norm. For example, a continuous function may be obtained as the point-wise, but not necessary uniform, limit of other continuous functions. Even though the limit is not a norm limit, one may still need to know that the norm of the limit function is no greater than the norms of the approximating functions. The classical case is, of course, Fatou’s lemma: if fn f pointwise, then

∫            ∫
|f| ≦ lim inf |f |.

Another common situation is this. A subspace A C(X) is given which has a norm, f sup|f|. If fn f pointwise (or uniformly), does it follow that f liminf fn? The answer is “yes” quite often, but can be “no,” even when A is a subalgebra.

Mathematical Subject Classification 2000
Primary: 46B99
Secondary: 46E25
Received: 18 February 1971
Published: 1 July 1971
Stephen Scheinberg