Vector-valued Hahn–Banach theorems

In mathematics, specifically in functional analysis and Hilbert space theory, vector-valued Hahn–Banach theorems are generalizations of the Hahn–Banach theorems from linear functionals (which are always valued in the real numbers $ℝ$ or the complex numbers $ℂ$) to linear operators valued in topological vector spaces (TVSs).

Definitions

Throughout Template:Mvar and Template:Mvar will be topological vector spaces (TVSs) over the field $𝕂$ and Template:Math will denote the vector space of all continuous linear maps from Template:Mvar to Template:Mvar, where if Template:Mvar and Template:Mvar are normed spaces then we endow Template:Math with its canonical operator norm.

Extensions

If Template:Mvar is a vector subspace of a TVS Template:Mvar then Template:Mvar has the extension property from Template:Mvar to Template:Mvar if every continuous linear map Template:Math has a continuous linear extension to all of Template:Mvar. If Template:Mvar and Template:Mvar are normed spaces, then we say that Template:Mvar has the metric extension property from Template:Mvar to Template:Mvar if this continuous linear extension can be chosen to have norm equal to Template:Math.

A TVS Template:Mvar has the extension property from all subspaces of Template:Mvar (to Template:Mvar) if for every vector subspace Template:Mvar of Template:Mvar, Template:Mvar has the extension property from Template:Mvar to Template:Mvar. If Template:Mvar and Template:Mvar are normed spaces then Template:Mvar has the metric extension property from all subspace of Template:Mvar (to Template:Mvar) if for every vector subspace Template:Mvar of Template:Mvar, Template:Mvar has the metric extension property from Template:Mvar to Template:Mvar.

A TVS Template:Mvar has the extension propertyTemplate:Sfn if for every locally convex space Template:Mvar and every vector subspace Template:Mvar of Template:Mvar, Template:Mvar has the extension property from Template:Mvar to Template:Mvar.

A Banach space Template:Mvar has the metric extension propertyTemplate:Sfn if for every Banach space Template:Mvar and every vector subspace Template:Mvar of Template:Mvar, Template:Mvar has the metric extension property from Template:Mvar to Template:Mvar.

1-extensions

If Template:Mvar is a vector subspace of normed space Template:Mvar over the field $𝕂$ then a normed space Template:Mvar has the immediate 1-extension property from Template:Mvar to Template:Mvar if for every Template:Math, every continuous linear map Template:Math has a continuous linear extension $F:M\oplus (𝕂x)\to Y$ such that Template:Math. We say that Template:Mvar has the immediate 1-extension property if Template:Mvar has the immediate 1-extension property from Template:Mvar to Template:Mvar for every Banach space Template:Mvar and every vector subspace Template:Mvar of Template:Mvar.

Injective spaces

A locally convex topological vector space Template:Mvar is injectiveTemplate:Sfn if for every locally convex space Template:Mvar containing Template:Mvar as a topological vector subspace, there exists a continuous projection from Template:Mvar onto Template:Mvar.

A Banach space Template:Mvar is 1-injectiveTemplate:Sfn or a Template:Math-space if for every Banach space Template:Mvar containing Template:Mvar as a normed vector subspace (i.e. the norm of Template:Mvar is identical to the usual restriction to Template:Mvar of Template:Mvar's norm), there exists a continuous projection from Template:Mvar onto Template:Mvar having norm 1.

Properties

In order for a TVS Template:Mvar to have the extension property, it must be complete (since it must be possible to extend the identity map $\mathrm{𝟏}:Y\to Y$ from Template:Mvar to the completion Template:Mvar of Template:Mvar; that is, to the map Template:Math).Template:Sfn

Existence

If Template:Math is a continuous linear map from a vector subspace Template:Mvar of Template:Mvar into a complete Hausdorff space Template:Mvar then there always exists a unique continuous linear extension of Template:Mvar from Template:Mvar to the closure of Template:Mvar in Template:Mvar.Template:Sfn Consequently, it suffices to only consider maps from closed vector subspaces into complete Hausdorff spaces.Template:Sfn

Results

Any locally convex space having the extension property is injective.Template:Sfn If Template:Mvar is an injective Banach space, then for every Banach space Template:Mvar, every continuous linear operator from a vector subspace of Template:Mvar into Template:Mvar has a continuous linear extension to all of Template:Mvar.Template:Sfn

In 1953, Alexander Grothendieck showed that any Banach space with the extension property is either finite-dimensional or else Template:Em separable.Template:Sfn

Examples

Products of the underlying field

Suppose that $X$ is a vector space over $𝕂$, where $𝕂$ is either $ℝ$ or $ℂ$ and let $T$ be any set. Let $Y:={𝕂}^T,$ which is the product of $𝕂$ taken $|T|$ times, or equivalently, the set of all $𝕂$-valued functions on Template:Mvar. Give $Y$ its usual product topology, which makes it into a Hausdorff locally convex TVS. Then $Y$ has the extension property.Template:Sfn

For any set $T,$ the Lp space ${\ell }^{\mathrm{\infty }}(T)$ has both the extension property and the metric extension property.

See also

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References

• Template:Narici Beckenstein Topological Vector Spaces
• Template:Rudin Walter Functional Analysis
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels