# 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 → 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 $\mathbb {K}$ 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 $F:M\oplus (\mathbb {K} x)\to Y$ 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 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 → 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 $\mathbb {K} .$ 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 — 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.
For any set $T,$ the Lp space $\ell ^{\infty }(T)$ has both the extension property and the metric extension property.