Hahn-Banach theorem
The Hahn-Banach theorem is a foundational result in functional analysis. Roughly speaking, it asserts the existence of a great variety of bounded (and hence continuous) linear functionals on an normed vector space, even if that space happens to be infinite-dimensional. We first consider an abstract version of this theorem, and then give the more classical result as a corollary.
Let be a real, or a complex vector space, with denoting the corresponding field of scalars, and let
be a seminorm on .
Theorem 1
Let be a linear functional defined on a subspace . If the restricted functional satisfies
then it can be extended to all of without violating the above property. To be more precise, there exists a linear functional such that
Definition 2
Theorem 3 (Hahn-Banach)
Let be a bounded linear functional defined on a subspace . Let denote the norm of relative to the restricted seminorm on . Then there exists a bounded extension 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 |