Hahn-Banach theorem

The Hahn-Banach theoremMathworldPlanetmath is a foundational result in functional analysisMathworldPlanetmathPlanetmath. Roughly speaking, it asserts the existence of a great varietyMathworldPlanetmath of bounded (and hence continuousMathworldPlanetmathPlanetmath) linear functionalsMathworldPlanetmathPlanetmath on an normed vector spacePlanetmathPlanetmath, even if that space happens to be infinite-dimensional. We first consider an abstract version of this theoremMathworldPlanetmath, and then give the more classical result as a corollary.

Let V be a real, or a complex vector space, with K denoting the corresponding field of scalars, and let


be a seminormMathworldPlanetmath on V.

Theorem 1

Let f:UK be a linear functional defined on a subspacePlanetmathPlanetmath UV. If the restricted functionalMathworldPlanetmathPlanetmath satisfies


then it can be extended to all of V without violating the above property. To be more precise, there exists a linear functional F:VK such that

F(𝐮) =f(𝐮),𝐮U
|F(𝐮)| p(𝐮),𝐮V.
Definition 2

We say that a linear functional f:VK is bounded if there exists a bound BR+ such that

|f(𝐮)|Bp(𝐮),𝐮V. (1)

If f is a bounded linear functional, we define f, the norm of f, according to


One can show that f is the infimumMathworldPlanetmath of all the possible B that satisfy (1)

Theorem 3 (Hahn-Banach)

Let f:UK be a bounded linear functional defined on a subspace UV. Let fU denote the norm of f relative to the restricted seminorm on U. Then there exists a bounded extensionPlanetmathPlanetmath F:VK with the same norm, i.e.

Title Hahn-Banach theorem
Canonical name HahnBanachTheorem
Date of creation 2013-03-22 12:54:09
Last modified on 2013-03-22 12:54:09
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 10
Author rmilson (146)
Entry type Theorem
Classification msc 46B20
Defines bound
Defines bounded