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 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.
|Date of creation||2013-03-22 12:54:09|
|Last modified on||2013-03-22 12:54:09|
|Last modified by||rmilson (146)|