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 $$\R$$ 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 separable.

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Examples
Products of the underlying field

Suppose that $$X$$ is a vector space over $$\mathbb{K}$$, where $$\mathbb{K}$$ is either $$\R$$ or $$\Complex$$ 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.