 Complex Hahn-Banach theorem

A bounded complex-linear linear functional on a complex linear subspace of a normed space extends to the whole space with the same operator norm. To deduce this from the real Hahn-Banach theorem, extend the real part $u$ under the bound $\Vert \ell \Vert \Vert x\Vert$ on the underlying real vector space, then put $L(x)=u(x)-iu(ix)$. This is complex-linear and agrees with $\ell$ on its original domain. Given $x$, choose a complex number $a$ of modulus one with $aL(x)=|L(x)|$. Then $|L(x)|=\mathrm{Re}L(ax)=u(ax)\le \Vert \ell \Vert \Vert x\Vert$. Restriction gives the reverse operator norm inequality.