PlanetMath: Hahn-Banach theorem

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Hahn-Banach theorem

(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

$\displaystyle \operatorname{p}:V\rightarrow\mathbb{R}^+$

be a seminorm on

Theorem 1 Let $f:U\to K$ be a linear functional defined on a subspace $U\subset V$ . If the restricted functional satisfies

$\displaystyle \vert f(\mathbf{u})\vert\leq \operatorname{p}(\mathbf{u}),\quad \mathbf{u}\in U,$

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

$\displaystyle F(\mathbf{u})$	$\displaystyle = f(\mathbf{u}),\quad \mathbf{u}\in U$
$\displaystyle \vert F(\mathbf{u}) \vert$	$\displaystyle \leq \operatorname{p}(\mathbf{u}),\quad \mathbf{u}\in V.$

Definition 2 We say that a linear functional $f:V\to K$ is bounded if there exists a bound $B\in\mathbb{R}^+$ such that

$\displaystyle \vert f(\mathbf{u})\vert \leq B \operatorname{p}(\mathbf{u}),\quad \mathbf{u}\in V.$

(1)

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

$\displaystyle \Vert f \Vert = \sup \{ \vert f(\mathbf{u})\vert : \operatorname{p}(\mathbf{u}) = 1 \}.$

One can show that $\Vert f\Vert$ is the infimum of all the possible that satisfy (1)

Theorem 3 (Hahn-Banach) Let $f:U\to K$ be a bounded linear functional defined on a subspace $U\subset V$ . Let $\Vert f \Vert_U$ denote the norm of relative to the restricted seminorm on . Then there exists a bounded extension $F:V\to K$ with the same norm, i.e.

$\displaystyle \Vert F\Vert_V = \Vert f\Vert_U.$

"Hahn-Banach theorem" is owned by rmilson. [ full author list (2) | owner history (1) ]

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Also defines:

bound, bounded

Attachments:: proof of Hahn-Banach theorem (Proof) by paolini

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Cross-references: extension, infimum, norm, property, functional, restricted, subspace, seminorm, scalars, field, vector space, complex, real, infinite-dimensional, even, normed vector space, linear functionals, continuous, variety, functional analysis
There are 67 references to this entry.

This is version 7 of Hahn-Banach theorem, born on 2002-08-01, modified 2003-04-13.
Object id is 3252, canonical name is HahnBanachTheorem.
Accessed 14245 times total.

Classification:

AMS MSC:

46B20 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Geometry and structure of normed linear spaces)

Pending Errata and Addenda

None.[ View all 2 ]

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