Hahn-Banach theorem

The Hahn-Banach theorem is the foundational result of functional analysis stating that a bounded linear functional defined on a subspace of a normed vector space can be extended to the entire space without increasing its norm. The theorem has multiple equivalent forms — geometric, analytic, and complex — and its power lies in its guarantee that normed spaces have "enough" continuous linear functionals to separate points and support a rich duality theory. Without Hahn-Banach, the dual space of a Banach space might be trivial, and the entire edifice of functional analysis would collapse. The standard proof uses Zorn's lemma, making the theorem equivalent to the axiom of choice in its full generality. The Hahn-Banach theorem is not a technical lemma; it is the statement that local constraints can be globalized, that what holds in a part can be made to hold in the whole without loss.