# bounded linear extension of an operator

## 0.1 Bounded Linear Extension

Let $X$ and $Y$ be normed vector spaces^{} and denote by $\stackrel{~}{X}$ and $\stackrel{~}{Y}$ their completions.

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Theorem 1 - Every bounded linear operator $T:X\u27f6Y$ can be extended to a bounded linear operator $\stackrel{~}{T}:\stackrel{~}{X}\u27f6\stackrel{~}{Y}$. Moreover, this extension^{} is unique and $\parallel T\parallel =\parallel \stackrel{~}{T}\parallel $.

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In particular, if $Y$ is a Banach space^{} and $S\subseteq X$ is a (not necessarily closed (http://planetmath.org/ClosedSet)) subspace^{} of $X$, an operator $T:S\u27f6Y$ has an extension $\stackrel{~}{T}:\overline{S}\u27f6Y$ to $\overline{S}$ (the closure^{} (http://planetmath.org/Closure) of $S$), which is unique and such that $\parallel T\parallel =\parallel \stackrel{~}{T}\parallel $.

## 0.2 Functorial Property of the Extension

The extension of bounded linear operators between two normed vector spaces to their completions is functorial. More precisely, let $\mathrm{\mathbf{N}\mathbf{V}\mathbf{e}\mathbf{c}}$ be the category^{} of normed vector spaces (whose morphisms^{} (http://planetmath.org/Category) are the bounded linear operators) and $\mathrm{\mathbf{B}\mathbf{a}\mathbf{n}}$ the categroy of Banach spaces (whose are also the bounded linear operators). We have that

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Theorem 2 - The completion $\stackrel{~}{}:\mathrm{\mathbf{N}\mathbf{V}\mathbf{e}\mathbf{c}}\u27f6\mathrm{\mathbf{B}\mathbf{a}\mathbf{n}}$, which associates each normed vector space $X$ with its completion $\stackrel{~}{X}$ and each bounded linear operator $T$ with its extension $\stackrel{~}{T}$, is a covariant functor^{}.

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This, in particular, implies that $\stackrel{~}{{T}_{1}{T}_{2}}=\stackrel{~}{{T}_{1}}\stackrel{~}{{T}_{2}}$.

## 0.3 Extensions in Spaces with Additional Structure

When the normed vector spaces $X$ and $Y$ have some additional structure^{} (for example, when $X$ and $Y$ are normed algebras) it is interesting to know if the (unique) extension of a morphism $T:X\u27f6Y$ preserves the additional structure. The following theorem states that this indeed the case for normed algebras or normed *-algebras^{}.

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Theorem 3 - If $X$ and $Y$ be normed vector spaces that are also normed algebras (normed *-algebras) and $T:X\u27f6Y$ is a bounded^{} homomorphism^{} (bounded *-homomorphism), then the unique bounded linear extension $\stackrel{~}{T}$ of $T$ is also an homomorphism (*-homomorphism).

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Thus, completion is also a covariant functor from the category of normed algebras (normed *-algebras) to category of Banach algebras^{} (Banach *-algebras).

Title | bounded linear extension of an operator |
---|---|

Canonical name | BoundedLinearExtensionOfAnOperator |

Date of creation | 2013-03-22 17:35:17 |

Last modified on | 2013-03-22 17:35:17 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 11 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46B99 |

Classification | msc 47A05 |

Synonym | continuous^{} extension of an operator |

Defines | completion of normed spaces is a covariant functor |

Defines | continuous extension of a normed algebra homomorphism |