Annihilating a subspace of 𝐿₁ with the sign of a continuous function

Wulbert, Daniel

doi:10.1090/S0002-9939-99-05317-4

Annihilating a subspace of $L_1$ with the sign of a continuous function
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Proc. Amer. Math. Soc. 128 (2000), 2431-2438 Request permission

Abstract:

Let $(X,\Sigma , \mu )$ be a $\sigma$-finite, nonatomic, Baire measure space. Let $G$ be a finite dimensional subspace of $L_1(X,\Sigma , \mu )$. There is a bounded, continuous function, $q$, defined on $X$, such that (1) $\int _X g\operatorname {sgn} q d\mu =0$ for all $g \in G$, and (2) $|\operatorname {sgn} q | =1$ almost everywhere.

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