proof of bounded linear functionals on Lp(μ)

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

For any gLq, define the linear map


This is a bounded linear map with operator norm Φg=gq (see Lp-norm is dual to Lq (, 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 (, 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


for every fL (see bounded linear functionalsMathworldPlanetmathPlanetmath on L ( Set g=h-1g0 and, for any fLp, let fn be the sequence


As h-1fnL,


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 (, so gLq. As |fng||fg| are in L1, dominated convergence finally gives


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