The **Hahn-Banach theorem** is a central tool in functional analysis; it shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space interesting ( though that is a matter of opinion ).

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":

- 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 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*x*is an element of_{0}*V*not in*U*, then there exists a continuous linear map ψ :*V*`->`**K**with ψ(*x*) = 0 for all*x*in*U*, ψ(*x*) = 1, and ||ψ|| = ||_{0}*x*||_{0}^{-1}.