# ${L}^{p}$-norm is dual to ${L}^{q}$

If $(X,\U0001d510,\mu )$ is any measure space^{} and $1\le p,q\le \mathrm{\infty}$ are Hölder conjugates (http://planetmath.org/ConjugateIndex) then, for $f\in {L}^{p}$, the following linear function can be defined

${\mathrm{\Phi}}_{f}:{L}^{q}\to \u2102,$ | ||

$g\mapsto {\mathrm{\Phi}}_{f}(g)\equiv {\displaystyle \int fg\mathit{d}\mu}.$ |

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

$$\parallel {\mathrm{\Phi}}_{f}\parallel =\{{\parallel fg\parallel}_{1}:g\in {L}^{q},{\parallel g\parallel}_{q}=1\}.$$ |

The following theorem shows that the operator norm of ${\mathrm{\Phi}}_{f}$ is equal to the ${L}^{p}$-norm of $f$.

###### Theorem.

Let $\mathrm{(}X\mathrm{,}\mathrm{M}\mathrm{,}\mu \mathrm{)}$ be a $\sigma $-finite measure space and $p\mathrm{,}q$ be Hölder conjugates. Then, any measurable function^{} $f\mathrm{:}X\mathrm{\to}\mathrm{C}$ has ${L}^{p}$-norm

$${\parallel f\parallel}_{p}=sup\{{\parallel fg\parallel}_{1}:g\in {L}^{q},{\parallel g\parallel}_{q}=1\}.$$ | (1) |

Furthermore, if either $$ and $$ or $p\mathrm{=}\mathrm{1}$ then $\mu $ is not required to be $\sigma $-finite.

Note that the $\sigma $-finite condition is required, except in the cases mentioned. For example, if $\mu $ is the measure satisfying $\mu (A)=\mathrm{\infty}$ for every nonempty set $A$, then ${L}^{p}(\mu )=\{0\}$ for $$ and it is easily checked that equality (1) fails whenever $f=1$ and $p>1$.

Title | ${L}^{p}$-norm is dual to ${L}^{q}$ |
---|---|

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 |