The most general formulation of the theorem needs some preparations. If V is a vector space over the scalar field K (either the real numbers R or the complex numbers C), we call a function N : V -> R sublinear if N(ax + by) ≤ |a| N(x) + |b| N(y) for all x and y in V and all scalars a and b in K. Every norm on V is sublinear, but there are other examples.
Now let U be a subspace of V and let φ : U -> K be a [linear function]? such that |φ(x)| ≤ N(x) for all x in U. Then the Hahn-Banach theorem states that there exists a linear map ψ : V -> K which extends φ (meaning ψ(x) = φ(x) for all x in U) and which is dominated by N on all of V (meaning |ψ(x)| ≤ N(x) for all x in V).
The extension ψ is in general not uniquely specified by φ and the proof gives no method as to how to find ψ: it depends on Zorn's lemma.
Several important consequences of the theorem are also sometimes called "Hahn-Banach theorem":