= 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 $\|\ell\|\|x\|$ 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 $|L(x)|=\operatorname{Re}L(ax)=u(ax)\leq\|\ell\|\|x\|$. Restriction gives the reverse <operator norm> inequality.
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