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

$$\mathrm{p}:V\to {\mathbb{R}}^{+}$$ |

be a seminorm^{} on $V$.

###### Theorem 1

Let $f\mathrm{:}U\mathrm{\to}K$ be a linear functional defined on a subspace^{}
$U\mathrm{\subset}V$. If the restricted functional^{} satisfies

$$|f(\mathbf{u})|\le \mathrm{p}(\mathbf{u}),\mathbf{u}\in U,$$ |

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

$F(\mathbf{u})$ | $=f(\mathbf{u}),\mathbf{u}\in U$ | ||

$|F(\mathbf{u})|$ | $\le \mathrm{p}(\mathbf{u}),\mathbf{u}\in V.$ |

###### Definition 2

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

$$|f(\mathbf{u})|\le B\mathrm{p}(\mathbf{u}),\mathbf{u}\in V.$$ | (1) |

If $f$ is a bounded linear functional, we define $\mathrm{\parallel}f\mathrm{\parallel}$, the norm of $f$, according to

$$\parallel f\parallel =sup\{|f(\mathbf{u})|:\mathrm{p}(\mathbf{u})=1\}.$$ |

One can show that $\mathrm{\parallel}f\mathrm{\parallel}$ is the infimum^{} of all the possible
$B$ that satisfy (1)

###### Theorem 3 (Hahn-Banach)

Let $f\mathrm{:}U\mathrm{\to}K$ be a bounded linear functional defined on a subspace
$U\mathrm{\subset}V$. Let ${\mathrm{\parallel}f\mathrm{\parallel}}_{U}$ denote the norm of $f$ relative
to the restricted seminorm on $U$. Then there exists a bounded
extension^{} $F\mathrm{:}V\mathrm{\to}K$ with the same norm, i.e.

$${\parallel F\parallel}_{V}={\parallel f\parallel}_{U}.$$ |

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 |