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

1 2 3 4 5 6 7 8 9 10 11 12 13 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.

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[FOOTNOTENariciBeckenstein2011341–370-1] 1 2 3 4 5 6 7 8 9 10 11 12 13 Narici & Beckenstein 2011, pp. 341–370.

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

[1]

[2]

Topological vector spaces (TVSs)
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