bounded linear functionals on Lp(μ)

If μ is a positive measureMathworldPlanetmath on a set X, 1p, and gLq(μ), where q is the Hölder conjugate of p, then Hölder’s inequalityMathworldPlanetmath implies that the map fXfg𝑑μ is a bounded linear functionalMathworldPlanetmath on Lp(μ). It is therefore natural to ask whether or not all such functionalsMathworldPlanetmathPlanetmath on Lp(μ) are of this form for some gLq(μ). Under fairly mild hypotheses, and excepting the case p=, the Radon-Nikodym TheoremMathworldPlanetmath answers this question affirmatively.


Let (X,M,μ) be a σ-finite measure space, 1p<, and q the Hölder conjugate of p. If Φ is a bounded linear functional on Lp(μ), then there exists a unique gLq(μ) such that

Φ(f)=Xfg𝑑μ (1)

for all fLp(μ). Furthermore, Φ=gq. Thus, under the stated hypotheses, Lq(μ) is isometrically isomorphic to the dual spaceMathworldPlanetmathPlanetmath of Lp(μ).

If 1<p<, then the assertion of the theorem remains valid without the assumptionPlanetmathPlanetmath that μ is σ-finite; however, even with this hypothesis, the result can fail in the case that p=. In particular, the bounded linear functionals on L(m), where m is Lebesgue measureMathworldPlanetmath on [0,1], are not all obtained in the above manner via members of L1(m). An explicit example illustrating this is constructed as follows: the assignment ff(0) defines a bounded linear functional on C([0,1]), which, by the Hahn-Banach Theorem, may be extended to a bounded linear functional Φ on L(m). Assume for the sake of contradictionMathworldPlanetmathPlanetmath that there exists gL1(m) such that Φ(f)=[0,1]fg𝑑m for every fL(m), and for n+, define fn:[0,1] by fn(x)=max{1-nx,0}. As each fn is continuousMathworldPlanetmathPlanetmath, we have Φ(fn)=φ(fn)=1 for all n; however, because fn0 almost everywhere and |fn|1, the Dominated Convergence Theorem, together with our hypothesis on g, gives


a contradiction. It follows that no such g can exist.

Title bounded linear functionals on Lp(μ)
Canonical name BoundedLinearFunctionalsOnLpmu
Date of creation 2013-03-22 18:32:57
Last modified on 2013-03-22 18:32:57
Owner azdbacks4234 (14155)
Last modified by azdbacks4234 (14155)
Numerical id 15
Author azdbacks4234 (14155)
Entry type Theorem
Classification msc 28B15
Related topic LpSpace
Related topic HolderInequality
Related topic ContinuousLinearMapping
Related topic BanachSpace
Related topic DualSpace
Related topic ConjugateIndex
Related topic RadonNikodymTheorem
Related topic BoundedLinearFunctionalsOnLinftymu
Related topic LpNormIsDualToLq