= Closed-hyperplane proof of Lp duality
{title2=$\ell(f)=\int fg$}
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 $g=|z|^{p-2}\overline z/\|z\|_p^p\in L^q$. <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.
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