bounded linear extension of an operator
0.1 Bounded Linear Extension
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 be the category of normed vector spaces (whose morphisms (http://planetmath.org/Category) are the bounded linear operators) and the categroy of Banach spaces (whose are also the bounded linear operators). We have that
Theorem 2 - The completion , which associates each normed vector space with its completion and each bounded linear operator with its extension , is a covariant functor.
This, in particular, implies that .
0.3 Extensions in Spaces with Additional Structure
When the normed vector spaces and have some additional structure (for example, when and are normed algebras) it is interesting to know if the (unique) extension of a morphism preserves the additional structure. The following theorem states that this indeed the case for normed algebras or normed *-algebras.
Theorem 3 - If and be normed vector spaces that are also normed algebras (normed *-algebras) and is a bounded homomorphism (bounded *-homomorphism), then the unique bounded linear extension of is also an homomorphism (*-homomorphism).
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|
|Date of creation||2013-03-22 17:35:17|
|Last modified on||2013-03-22 17:35:17|
|Last modified by||asteroid (17536)|
|Synonym||continuous extension of an operator|
|Defines||completion of normed spaces is a covariant functor|
|Defines||continuous extension of a normed algebra homomorphism|