Hahn–Banach theorem
The Hahn–Banach theorem guarantees that there are always enough continuous functionals to do geometry. Whenever you have measured vectors on a small subspace in a controlled way, you can extend that measurement to the whole space without ever exceeding the control. It is a promise that the dual space is large and rich — large enough to separate points, support hyperplanes, and detect directions.
In its normed-space form: let Y be a subspace of a normed space X and let f be a bounded linear functional on Y. Then there exists a bounded linear functional F on all of X that agrees with f on Y and has the same norm, ||F|| = ||f||. No enlargement of the domain forces any growth of the norm. (The more general analytic version replaces the norm by an arbitrary sublinear functional p and only requires F <= p.)
Its consequences are everywhere. For any nonzero x there is a unit-norm functional with f(x) = ||x||, so functionals genuinely see every vector; points and closed subspaces can be separated by hyperplanes; and the canonical embedding of X into its double dual is an isometry. Unlike the deeper Banach-space theorems, Hahn–Banach needs no completeness — but in full generality it does rely on the axiom of choice (via Zorn's lemma).
On the subspace Y = { (t, 0) : t real } of R^2 with the max norm, define f(t, 0) = t, so ||f|| = 1. Extensions to R^2 of the form F(x, y) = x + c y keep norm 1 precisely when |c| <= 1, giving infinitely many norm-preserving extensions — a finite-dimensional snapshot of the non-uniqueness Hahn–Banach permits.
Many distinct norm-preserving extensions of one functional, illustrating non-uniqueness.
The extension is generally not unique — there can be many norm-preserving extensions — and the theorem asserts existence, not a formula. Over the complex field one first handles the real part and then reconstructs the complex functional, a standard two-step technique worth remembering.