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

Let $(X,\mathfrak{M},\mu)$ be a $\sigma$-finite measure space and $p,q$ be Hölder conjugates. Then, we show that a measurable function $f\colon X\rightarrow\mathbb{R}$ has $L^{p}$-norm

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

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

If $\|f\|_{p}=0$ then $f$ is zero almost everywhere, and both sides of equality (1) are zero. So, we only need to consider the case where $\|f\|_{p}>0$.

Let $K$ be the right hand side of equality (1). For any $g\in L^{q}$ with $\|g\|_{q}=1$, the Hölder inequality gives $\|fg\|_{1}\leq\|f\|_{p}$, so $K\leq\|f\|_{p}$. Only the reverse inequality remains to be shown.

If $1 and $\|f\|_{p}<\infty$ then, setting $g=|f|^{p-1}$ gives

 $\|g\|_{q}=\left(\int|f|^{p}\,d\mu\right)^{\frac{1}{q}}=\|f\|_{p}^{p-1}<\infty.$

Therefore, $g\in L^{q}$ and,

 $K\geq\|f(g/\|g\|_{q})\|_{1}=\||f|^{p}\|_{1}/\|g\|_{q}=\|f\|_{p}^{p}/\|f\|_{p}^% {p-1}=\|f\|_{p}.$

On the other hand, if $p=1$ so that $q=\infty$, then setting $g=1$ gives $\|g\|_{q}=1$ and

 $K\geq\|fg\|_{1}=\|f\|_{1}.$

So, we have shown that $K=\|f\|_{p}$ when $p<\infty$ and $\|f\|_{p}<\infty$, and when $p=1$. From now on, it is assumed that the measure is $\sigma$-finite. Then there is a sequence $A_{n}\in\mathfrak{M}$ increasing to the whole of $X$ and such that $\mu(A_{n})<\infty$.

Now consider the case where $1 and $\|f\|_{p}=\infty$. Let $f_{n}$ be the sequence of functions

 $f_{n}=1_{A_{n}}1_{|f|\leq n}f$

then, $|f_{n}|\leq|f|$ and monotone convergence gives $\|f_{n}\|_{p}\rightarrow\|f\|_{p}=\infty$. Therefore,

 $K\geq\sup\left\{\|f_{n}g\|_{1}:g\in L^{q},\|g\|_{q}=1\right\}=\|f_{n}\|_{p}.$

and letting $n$ go to infinity gives $K=\infty$.

We finally consider $p=\infty$. Then, for any $L<\|f\|_{p}$ there exists a set $A\in\mathfrak{M}$ with $\mu(A)>0$ such that $|f|\geq L$ on $A$. Also, monotone convergence gives $\mu(A\cap A_{n})\rightarrow\mu(A)$ and, therefore, $\mu(A\cap A_{n})>0$ eventually. Replacing $A$ by $A\cap A_{n}$ if necessary, we may suppose that $\mu(A)<\infty$. So, setting $g=1_{A}/\mu(A)$ gives $\|g\|_{1}=1$ and,

 $K\geq\|fg\|_{1}=\int_{A}|f|\,d\mu/\mu(A)\geq L.$

Letting $L$ increase to $\|f\|_{p}$ gives $K\geq\|f\|_{p}$ as required.

Title proof of $L^{p}$-norm is dual to $L^{q}$ ProofOfLpnormIsDualToLq 2013-03-22 18:38:16 2013-03-22 18:38:16 gel (22282) gel (22282) 4 gel (22282) Proof msc 46E30 msc 28A25