Closed-hyperplane proof of Lp duality ($\ell (f)=\int fg$)

Let $1<p<\infty$ and $q=p/(p-1)$. For a nonzero bounded linear functional $\ell$ on an Lp space, choose $f_0$ with $\ell (f_0)=1$ and a closest point $h$ in its closed kernel. Put $z=f_0-h$. The variational condition is $\int |z{|}^{p-2}\overline{z}k=0$ for every kernel element $k$. Since $f-\ell (f)z$ lies in the kernel, $\ell (f)=\int fg$ with $g=|z{|}^{p-2}\overline{z}/\Vert z{\Vert }_p^p\in L^q$. Hölder's inequality gives the upper norm bound; testing against $|g{|}^{q-2}\overline{g}$ gives equality. Thus the representation is an isometric linear isomorphism when the pairing is bilinear.