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.

The Hahn-Banach theorem states that:

- Let
*N*:*V*`->`**R**be sublinear, let*U*be a subspace of*V*and let φ :*U*`->`**K**be a linear functional such that |φ(*x*)| ≤*N*(*x*) for all*x*in*U*. Then 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*).

Several important consequences of the theorem are also sometimes called "Hahn-Banach theorem":

- If
*V*is a normed vector space with subspace*U*(not necessarily closed) and if φ :*U*`->`**K**is continuous and linear, then there exists an extension ψ :*V*`->`**K**of φ which is also continuous and linear and which has the same norm as φ (see Banach space for a discussion of the norm of a linear map). - If
*V*is a normed vector space with subspace*U*(not necessarily closed) and if*z*is an element of*V*not in the closure of*U*, then there exists a continuous linear map ψ :*V*`->`**K**with ψ(*x*) = 0 for all*x*in*U*, ψ(*z*) = 1, and ||ψ|| = ||*z*||^{-1}.

Lawrence Narici and Edward Beckenstein, 'The Hahn-Banach Theorem: The Life and Times', *Topology and its Applications*, Volume 77, Issue 2 (3 June 1997) Pages 193-211. An on-line preprint is available here