Dual pair on $L^p$

Question

I know that norms on $L^p(\mathbb{R}^n)$ spaces ($1<p<\infty$) are defined as $$\|f\|=\left(\int_{\mathbb{R}^n } |f|^p d\mu\right)^{1/p}.$$ But, is it possible to express the norm of $f$ using the dual pair $\langle\cdot,\cdot\rangle:L^p\times L^q\rightarrow \mathbb{R}$ where $p,q$ are conjugate exponents?

Answer 1

Yes, in fact you can do it explicitly. Simply find $g$ with $\| g \|_{L^q} = 1$ such that $fg = C|f|^p$, hence $\int f g = C \| f \|_{L^p}^p$. Then check that the requirement $\| g \|_{L^q}=1$ sets $C=\frac{1}{\| f \|_{L^p}^{p-1}}$.

In the special case $p=2$, you have the intuitive result $g=\frac{f}{\| f \|_{L^2}}$.

Answer 2

Yes, this is usually known as Riesz representation theorem. We have that for any $\omega \in (L^p)^*$ there is a unique $g\in L^q$ such that $\|\omega\|=\|g\|_q$. That is $\|g\|_q=\sup_{\|f\|_p=1}|\omega(f)|$; but under the dual pairing $|\omega(f)|=|\langle f, \omega \rangle |= |\langle f, g \rangle |$.