Duality of Lp spaces

Remarks

By Hölder's inequality, $\abs{\Phi(g)(f)} \le \norm{f}_p \norm{g}_q$, so $\Phi(g)$ is bounded with norm at most $\norm{g}_q$. The test function $f = \abs{g}^{q-1} \overline{\sgn g}$ gives equality when $1 < p < \infty$. Surjectivity is the hard part: for σ-finite $\mu$ it follows from the Radon–Nikodym theorem, and for $1 < p < \infty$ uniform convexity of $L^p$ removes the σ-finiteness.

For counting measure on $\N$, the theorem says that the dual of $\ell^p$ is $\ell^q$, and the dual of $\ell^1$ is $\ell^\infty$. The case $p = \infty$ is excluded: the dual of $\ell^\infty$ is strictly larger than $\ell^1$. For example, a Banach limit is a bounded functional on $\ell^\infty$ that is not given by any sequence in $\ell^1$. For $p = 1$, the σ-finite hypothesis cannot be dropped in general.

Going deeper

For a student

The theorem shows that $L^p$ is reflexive for $1 < p < \infty$, since the dual of $L^q$ is again $L^p$. In the case $p = 2$ it is a special case of the Riesz representation theorem for Hilbert spaces. It is used to identify weak limits, to define weak solutions of equations and to prove inequalities by duality, where one computes $\norm{f}_p$ as a supremum of $\abs{\int f g}$ over $\norm{g}_q \le 1$.