bounded linear extension of an operator

0.1 Bounded Linear Extension

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

$$

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

$$

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 $T:S⟶Y$ has an extension $\tilde{T}:\overline{S}⟶Y$ to $\overline{S}$ (the closure (http://planetmath.org/Closure) of $S$), which is unique and such that $\parallel T\parallel =\parallel \tilde{T}\parallel$.

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 $\mathrm{𝐍𝐕𝐞𝐜}$ be the category of normed vector spaces (whose morphisms (http://planetmath.org/Category) are the bounded linear operators) and $\mathrm{𝐁𝐚𝐧}$ the categroy of Banach spaces (whose are also the bounded linear operators). We have that

$$

Theorem 2 - The completion $\stackrel{~}{}:\mathrm{𝐍𝐕𝐞𝐜}⟶\mathrm{𝐁𝐚𝐧}$, which associates each normed vector space $X$ with its completion $\tilde{X}$ and each bounded linear operator $T$ with its extension $\tilde{T}$, is a covariant functor.

$$

This, in particular, implies that $\widetilde{T_1{T}_2}=\widetilde{T_1}\widetilde{T_2}$.

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 $T:X⟶Y$ preserves the additional structure. The following theorem states that this indeed the case for normed algebras or normed *-algebras.

$$

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