The Hahn-Banach Theorem
${ X \in \mathbb{R}\mbox{-}\mathsf{NVS}}$, ${ M \le X }$, sublinear functional ${ p }$ on ${ X }$, ${ f \in L(M,\mathbb{R}) }$ s.t. ${ f \le p }$
\[\exists F \in L(X,\mathbb{R}), \quad F \le p \mbox{ and } F\rvert_{M} = f\]The Complex Hahn-Banach Theorem
${ X \in \mathbb{C}\mbox{-}\mathsf{NVS}, M \le X }$, seminorm ${ p }$ on ${ X }$, ${ f \in L(M,\mathbb{C}) }$ s.t. ${ \lvert f \rvert \le p }$
\[\exists F \in L(X,\mathbb{C}), \quad \lvert F \rvert \le p \mbox{ and } F\rvert_{M}=f\]References
- Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Wiley, 1999.