proof of bounded linear functionals on Lp(μ)


If (X,𝔐,μ) is a σ-finite measure-space and p,q are Hölder conjugates (http://planetmath.org/ConjugateIndex) with p<, then we show that Lq is isometrically isomorphic to the dual spaceMathworldPlanetmathPlanetmathPlanetmath of Lp.

For any gLq, define the linear map

Φg:Lp,fΦg(f)=fg𝑑μ.

This is a bounded linear map with operator norm Φg=gq (see Lp-norm is dual to Lq (http://planetmath.org/LpNormIsDualToLq)), so the map gΦg gives an isometric embedding from Lq to the dual space of Lp. It only remains to show that it is onto.

So, suppose that Φ:Lp is a bounded linear map. It needs to be shown that there is a gLq with Φ=Φg. As any σ-finite measure is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to a probability measure (http://planetmath.org/AnySigmaFiniteMeasureIsEquivalentToAProbabilityMeasure), there is a boundedPlanetmathPlanetmathPlanetmathPlanetmath h>0 such that h𝑑μ=1. Let Φ~:L be the bounded linear map given by Φ~(f)=Φ(hf). Then, there is a g0L1 such that

Φ(hf)=Φ~(f)=fg0𝑑μ

for every fL (see bounded linear functionalsMathworldPlanetmathPlanetmath on L (http://planetmath.org/BoundedLinearFunctionalsOnLinftymu)). Set g=h-1g0 and, for any fLp, let fn be the sequence

fn=f1{|h-1f|<n}.

As h-1fnL,

fng1=h-1fng01=Φ(sign(fg0)fn)Φfnp.

Letting n tend to infinityMathworldPlanetmath, dominated convergence says that fnf in the Lp-norm, so Fatou’s lemma gives

fg1lim infnfng1Φfp.

In particular, gqΦ (see Lp-norm is dual to Lq (http://planetmath.org/LpNormIsDualToLq)), so gLq. As |fng||fg| are in L1, dominated convergence finally gives

fg𝑑μ=limnfng𝑑μ=limnΦ(fn)=Φ(f)

so Φg=Φ as required.

Title proof of bounded linear functionals on Lp(μ)
Canonical name ProofOfBoundedLinearFunctionalsOnLpmu
Date of creation 2013-03-22 18:38:19
Last modified on 2013-03-22 18:38:19
Owner gel (22282)
Last modified by gel (22282)
Numerical id 4
Author gel (22282)
Entry type Proof
Classification msc 46E30
Classification msc 28A25