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

Definitions

Throughout $X$ and $Y$ will be topological vector spaces (TVSs) over the field $\mathbb {K}$ 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 $f : M \to Y$ 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^[1] 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^[1] 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 $\mathbb {K}$ then a normed space $Y$ has the immediate 1-extension property from $M$ to $X$ if for every $x \notin M$ , every continuous linear map $f : M \to Y$ has a continuous linear extension $F:M\oplus (\mathbb {K} x)\to Y$ 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^[1] 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^[1] or a $P 1$ -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 $\mathbf {1} :Y\to Y$ from $Y$ to the completion $Z$ of $Y$ ; that is, to the map $Z \to Y$ ).^[1]

Existence

If $f : M \to Y$ 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 $X$ .^[1]^[2] Consequently, it suffices to only consider maps from closed vector subspaces into complete Hausdorff spaces.^[1]

Results

Any locally convex space having the extension property is injective.^[1] 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 $X$ .^[1]

In 1953, Alexander Grothendieck showed that any Banach space with the extension property is either finite-dimensional or else not separable.^[1]

Theorem^[1] — Suppose that $Y$ is a Banach space over the field $\mathbb {K} .$ Then the following are equivalent:

$Y$ is 1-injective;
$Y$ has the metric extension property;
$Y$ has the immediate 1-extension property;
$Y$ has the center-radius property;
$Y$ has the weak intersection property;
$Y$ is 1-complemented in any Banach space into which it is norm embedded;
Whenever $Y$ in norm-embedded into a Banach space $X$ then identity map $\mathbf {1} :Y\to Y$ can be extended to a continuous linear map of norm $1$ to $X$ ;
$Y$ is linearly isometric to $C\left(T,\mathbb {K} ,\|{\dot {}}\|_{\infty }\right)$ 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:

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

Theorem^[1] — Suppose that $Y$ is a real Banach space with the metric extension property. Then the following are equivalent:

$Y$ is reflexive;
$Y$ is separable;
$Y$ is finite-dimensional;
$Y$ is linearly isometric to $C\left(T,\mathbb {K} ,\|\cdot \|_{\infty }\right),$ for some discrete finite space $T.$

Examples

Products of the underlying field

Suppose that $X$ is a vector space over $\mathbb {K}$ , where $\mathbb {K}$ is either $\mathbb {R}$ or $\mathbb {C}$ and let $T$ be any set. Let $Y:=\mathbb {K} ^{T},$ which is the product of $\mathbb {K}$ taken $|T|$ times, or equivalently, the set of all $\mathbb {K}$ -valued functions on $T$ . Give $Y$ its usual product topology, which makes it into a Hausdorff locally convex TVS. Then $Y$ has the extension property.^[1]

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

Citations

^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m Narici & Beckenstein 2011, pp. 341–370.
^ 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) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

[FOOTNOTENariciBeckenstein2011341–370-1] ^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m Narici & Beckenstein 2011, pp. 341–370.

[2] Rudin 1991, p. 40 Stated for linear maps into F-spaces only; outlines proof.

[1]

[2]

v t e Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
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