Bounded Linear Extension

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

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

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In particular, if $Y$ is a Banach space and $S \subseteq X$ is a (not necessarily closed) subspace of $X$ , an operator $T:S \longrightarrow Y$ has an extension $\widetilde{T} : \overline{S} \longrightarrow Y$ to $\overline{S}$ (the closure of $S$ ), which is unique and such that $\|T\|= \|\widetilde{T}\|$ .

Functorial Property of the Extension

The extension of bounded linear operators between two normed vector spaces to their completions is functorial. More precisely, let $\mathbf{NVec}$ be the category of normed vector spaces (whose morphisms are the bounded linear operators) and $\mathbf{Ban}$ the categroy of Banach spaces (whose morphisms are also the bounded linear operators). We have that

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

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This, in particular, implies that $\widetilde{T_1T_2} = \widetilde{T_1}\widetilde{T_2}$ .

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 $T:X \longrightarrow Y$ 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 $T:X \longrightarrow Y$ is a bounded homomorphism (bounded *-homomorphism), then the unique bounded linear extension $\widetilde{T}$ 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).