Lp-norm is dual to Lq

If (X,𝔐,μ) is any measure spaceMathworldPlanetmath and 1p,q are Hölder conjugates (http://planetmath.org/ConjugateIndex) then, for fLp, the following linear function can be defined


The Hölder inequality (http://planetmath.org/HolderInequality) shows that this gives a well defined and bounded linear map. Its operator norm is given by


The following theorem shows that the operator norm of Φf is equal to the Lp-norm of f.


Let (X,M,μ) be a σ-finite measure space and p,q be Hölder conjugates. Then, any measurable functionMathworldPlanetmath f:XC has Lp-norm

fp=sup{fg1:gLq,gq=1}. (1)

Furthermore, if either p< and fp< or p=1 then μ is not required to be σ-finite.

Note that the σ-finite condition is required, except in the cases mentioned. For example, if μ is the measure satisfying μ(A)= for every nonempty set A, then Lp(μ)={0} for p< and it is easily checked that equality (1) fails whenever f=1 and p>1.

Title Lp-norm is dual to Lq
Canonical name LpnormIsDualToLq
Date of creation 2013-03-22 18:38:13
Last modified on 2013-03-22 18:38:13
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Theorem
Classification msc 28A25
Classification msc 46E30
Related topic LpSpace
Related topic HolderInequality
Related topic BoundedLinearFunctionalsOnLinftymu
Related topic BoundedLinearFunctionalsOnLpmu