Hahn-Banach is the mechanism that creates continuous linear functionals with prescribed behavior. The extension theorem is algebraic in form, but its analytic force appears when the dominating sublinear function comes from a norm or a convex set.
Sublinear domination
The extension problem is easiest when allowable values are encoded by a sublinear majorant. The one-dimensional step is the only real inequality in the proof.
Definition. A function \(p:X\to\mathbb R\) on a real vector space is sublinear if \(p(x+y)\le p(x)+p(y)\) and \(p(\lambda x)=\lambda p(x)\) for every \(\lambda\ge0\). A linear functional \(f\) on \(M\subset X\) is dominated by \(p\) if \(f(m)\le p(m)\) for all \(m\in M\).
Lemma. Let \(M\subset X\) be a subspace, \(x_0\notin M\), and \(f:M\to\mathbb R\) linear with \(f\le p\). There exists \(\alpha\in\mathbb R\) such that \(F(m+tx_0)=f(m)+t\alpha\) defines a linear extension to \(M+\mathbb Rx_0\) still satisfying \(F\le p\).
Proof. For \(m_1,m_2\in M\), sublinearity gives \(f(m_1)+f(m_2)=f(m_1+m_2)\le p(m_1+m_2)\le p(m_1+x_0)+p(m_2-x_0)\). Hence \(f(m_1)-p(m_1+x_0)\le p(m_2-x_0)-f(m_2)\).
Therefore the interval \([\sup_m(f(m)-p(m+x_0)),\inf_m(p(m-x_0)-f(m))]\) is nonempty. Choose \(\alpha\) in it. The inequalities \(\alpha\ge f(m)-p(m+x_0)\) and \(\alpha\le p(m-x_0)-f(m)\) are exactly the cases \(t>0\) and \(t<0\), after rescaling, needed to prove \(f(m)+t\alpha\le p(m+tx_0)\).
Hahn-Banach extension theorem
After the one-dimensional extension is known, the full theorem is a maximal-extension argument. Zorn's lemma handles the bookkeeping over all intermediate domains.
Theorem. Let \(p:X\to\mathbb R\) be sublinear on a real vector space and let \(f:M\to\mathbb R\) be linear on a subspace \(M\) with \(f\le p\). Then there is a linear \(F:X\to\mathbb R\) extending \(f\) such that \(F\le p\) on \(X\).
Proof. Partially order all dominated linear extensions \((N,g)\), where \(M\subset N\subset X\), by extension of domains. A chain has an upper bound obtained by taking the union of the domains and defining the functional consistently on that union. By Zorn's lemma there is a maximal dominated extension \((N_0,F_0)\).
If \(N_0\ne X\), choose \(x_0\notin N_0\). The one-dimensional extension lemma extends \(F_0\) to \(N_0+\mathbb Rx_0\) without losing domination, contradicting maximality. Hence \(N_0=X\).
Norm-preserving extension
Choosing the sublinear function from the norm turns the algebraic theorem into an analytic extension theorem for continuous functionals.
Corollary. If \(M\subset X\) is a subspace of a real normed space and \(f\in M^*\), then there exists \(F\in X^*\) extending \(f\) with \(\|F\|=\|f\|\).
Proof. Apply Hahn-Banach to \(p(x)=\|f\|\,\|x\|\). Since \(f(m)\le |f(m)|\le\|f\|\|m\|\), the extension \(F\) satisfies \(F(x)\le\|f\|\|x\|\). Applying this to \(-x\) gives \(|F(x)|\le\|f\|\|x\|\), so \(\|F\|\le\|f\|\). The reverse inequality holds because \(F\) extends \(f\).
Point separation and norm recovery
Point separation is the first major consequence: every nonzero vector can be detected by a continuous linear functional of controlled norm.
Corollary. For every \(x_0\ne0\) in a normed space \(X\), there exists \(f\in X^*\) such that \(\|f\|=1\) and \(f(x_0)=\|x_0\|\). Consequently \(\|x\|=\sup_{\|f\|\le1}|f(x)|\).
Proof. On \(M=\mathbb F x_0\), define \(f(\alpha x_0)=\alpha\|x_0\|\) in the real case, and \(f(\alpha x_0)=\alpha\|x_0\|\) in the complex case. This functional has norm \(1\). Extend it norm-preservingly. The inequality \(\sup_{\|f\|\le1}|f(x)|\le\|x\|\) is immediate; the separating functional gives equality for each fixed \(x\).
Complex Hahn-Banach
The complex case is recovered from the real theorem by extending the real part and reconstructing complex linearity.
Theorem. The norm-preserving extension theorem also holds for complex normed spaces.
Proof. Given a complex linear \(f\), let \(u=\operatorname{Re}f\). Then \(u\) is real linear and \(|u(m)|\le\|f\|\|m\|\). Extend \(u\) as a real linear functional \(U\) with the same norm by the real theorem. Define \(F(x)=U(x)-iU(ix)\). Then \(F\) is complex linear, extends \(f\), and \(\operatorname{Re}F=U\).
To control the norm, for \(F(x)\ne0\) choose \(\theta\) such that \(e^{i\theta}F(x)=|F(x)|\). Then \(|F(x)|=\operatorname{Re}F(e^{i\theta}x)=U(e^{i\theta}x)\le\|f\|\|x\|\). Thus \(\|F\|\le\|f\|\), and equality follows from extension.
Dual abundance
This final observation records the conceptual role of Hahn-Banach. The theorem guarantees enough continuous linear functionals for weak topologies and separation arguments to work.
Remark. Hahn-Banach is the theorem that prevents the continuous dual from being too small. Weak topologies, convex separation, norm recovery, adjoints, and many minimax arguments all depend on this abundance of continuous linear functionals.