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

Definitions

Throughout X and Y will be topological vector spaces (TVSs) over the field ${\displaystyle \mathbb {K} }$ 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 ${\displaystyle \mathbb {K} }$ 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 ${\displaystyle F:M\oplus (\mathbb {K} x)\to Y}$ 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 ${\displaystyle \mathbf {1} :Y\to Y}$ 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 ${\displaystyle \mathbb {K} .}$ 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 ${\displaystyle X}$ then identity map ${\displaystyle \mathbf {1} :Y\to Y}$ can be extended to a continuous linear map of norm ${\displaystyle 1}$ to ${\displaystyle X}$;
8. Y is linearly isometric to ${\displaystyle C\left(T,\mathbb {K} ,\|{\dot {}}\|_{\infty }\right)}$ 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 ${\displaystyle C\left(T,\mathbb {K} ,\|\cdot \|_{\infty }\right),}$ for some discrete finite space ${\displaystyle T.}$

Examples

Products of the underlying field

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

For any set ${\displaystyle T,}$ the Lp space ${\displaystyle \ell ^{\infty }(T)}$ 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

Citations

1. ^ Narici & Beckenstein 2011, pp. 341–370.
2. Rudin 1991, p. 40 Stated for linear maps into F-spaces only; outlines proof.

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 (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) . Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

• Topological vector spaces
• Theorems in functional analysis