The abstract norming-functional theorem agrees with the concrete L^p-L^q formula

Using the assumed HB, apply [L1] to the nonzero vector $f\in L^p(\mu )$. This gives a functional $\mathrm{\Phi }\in (L^p(\mu ){)}^{\ast }$ with $\mathrm{\parallel }\mathrm{\Phi }\mathrm{\parallel }=1$ and $\mathrm{\Phi }(f)=\mathrm{\parallel }f{\mathrm{\parallel }}_p$.

By [L2], the same number $\mathrm{\parallel }f{\mathrm{\parallel }}_p$ is obtained as $\sup \{\mid \int fgd\mu \mid :\mathrm{\parallel }g{\mathrm{\parallel }}_q\le 1\}\mathrm{.}$ For a representative $u$ of $f$, put $\theta (x)=\overline{u(x)}\mathrm{/}\mathrm{\mid }u(x)\mathrm{\mid }$ where $u(x)\mathrm{\ne }0$, and $\theta (x)=0$ otherwise. When $1<p<\mathrm{\infty }$, take $g=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}\theta \mathrm{/}\mathrm{\parallel }f{\mathrm{\parallel }}_p^{p-1}$. Then $\mathrm{\parallel }g{\mathrm{\parallel }}_q=1$ and $ug=\mathrm{\mid }u{\mathrm{\mid }}^p\mathrm{/}\mathrm{\parallel }f{\mathrm{\parallel }}_p^{p-1}$, so $\int ugd\mu =\mathrm{\parallel }f{\mathrm{\parallel }}_p$. When $p=1$, take $g=\theta$; then $\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$ and $ug=\mathrm{\mid }u\mathrm{\mid }$. The conjugate phase is required for the bilinear pairing $\int fg$ over complex scalars; over real scalars it is the usual sign.

By [L3], each displayed $g$ defines a functional of norm at most one, and its value on the nonzero $f$ is $\mathrm{\parallel }f{\mathrm{\parallel }}_p$, forcing its operator norm to be exactly one. Thus the abstract existence statement from Hahn-Banach and the concrete $L^p-L^q$ formula from the earlier page identify the same norming phenomenon: one proves that some unit functional attains $\mathrm{\parallel }f{\mathrm{\parallel }}_p$, and the other writes an attaining functional down explicitly.