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 [math]\displaystyle{ \R }[/math] or the complex numbers [math]\displaystyle{ \mathbb{C} }[/math]) to linear operators valued in topological vector spaces (TVSs).

Definitions

Throughout X and Y will be topological vector spaces (TVSs) over the field [math]\displaystyle{ \mathbb{K} }[/math] and L(X; Y) will denote the vector space of all continuous linear maps from X to Y, where if X and Y are normed spaces then we endow L(X; Y) with its canonical operator norm.

Extensions

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

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

A TVS Y has the extension property if for every locally convex space X and every vector subspace M of X, Y has the extension property from M to X.

A Banach space Y has the metric extension property if for every Banach space X and every vector subspace M of X, Y has the metric extension property from M to X.

1-extensions

If M is a vector subspace of normed space X over the field [math]\displaystyle{ \mathbb{K} }[/math] then a normed space Y has the immediate 1-extension property from M to X if for every x ∉ M, every continuous linear map f : M → Y has a continuous linear extension [math]\displaystyle{ F : M \oplus (\mathbb{K} x) \to Y }[/math] such that ||f|| = ||F||. We say that Y has the immediate 1-extension property if Y has the immediate 1-extension property from M to X for every Banach space X and every vector subspace M of X.

Injective spaces

A locally convex topological vector space Y is injective if for every locally convex space Z containing Y as a topological vector subspace, there exists a continuous projection from Z onto Y.

A Banach space Y is 1-injective or a P₁-space if for every Banach space Z containing Y as a normed vector subspace (i.e. the norm of Y is identical to the usual restriction to Y of Z's norm), there exists a continuous projection from Z onto Y having norm 1.

Properties

In order for a TVS Y to have the extension property, it must be complete (since it must be possible to extend the identity map [math]\displaystyle{ \mathbf{1} : Y \to Y }[/math] from Y to the completion Z of Y; that is, to the map Z → Y).

Existence

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

Results

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

In 1953, Alexander Grothendieck showed that any Banach space with the extension property is either finite-dimensional or else not separable.

Theorem — Suppose that Y is a Banach space over the field [math]\displaystyle{ \mathbb{K}. }[/math] Then the following are equivalent:

1. Y is 1-injective;
2. Y has the metric extension property;
3. Y has the immediate 1-extension property;
4. Y has the center-radius property;
5. Y has the weak intersection property;
6. Y is 1-complemented in any Banach space into which it is norm embedded;
7. Whenever Y in norm-embedded into a Banach space [math]\displaystyle{ X }[/math] then identity map [math]\displaystyle{ \mathbf{1} : Y \to Y }[/math] can be extended to a continuous linear map of norm [math]\displaystyle{ 1 }[/math] to [math]\displaystyle{ X }[/math];
8. Y is linearly isometric to [math]\displaystyle{ C\left(T, \mathbb{K}, \|\dot{}\|_{\infty}\right) }[/math] for some compact, Hausdorff space, extremally disconnected space T. (This space T is unique up to homeomorphism).

where if in addition, Y is a vector space over the real numbers then we may add to this list:

9. Y has the binary intersection property;
10. Y is linearly isometric to a complete Archimedean ordered vector lattice with order unit and endowed with the order unit norm.

Theorem — Suppose that Y is a real Banach space with the metric extension property. Then the following are equivalent:

1. Y is reflexive;
2. Y is separable;
3. Y is finite-dimensional;
4. Y is linearly isometric to [math]\displaystyle{ C\left(T, \mathbb{K}, \|\cdot\|_{\infty}\right), }[/math] for some discrete finite space [math]\displaystyle{ T. }[/math]

Examples

Products of the underlying field

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

For any set [math]\displaystyle{ T, }[/math] the Lp space [math]\displaystyle{ \ell^{\infty}(T) }[/math] has both the extension property and the metric extension property.

See also

• Continuous linear extension – Mathematical method in functional analysis
• Continuous linear operator
• Hahn–Banach theorem – Theorem on extension of bounded linear functionals
• Hyperplane separation theorem – On the existence of hyperplanes separating disjoint convex sets

References

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.