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 {\displaystyle \mathbb {R} } or the complex numbers C {\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 K {\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 K {\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 F : M ⊕ ( K x ) → Y {\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 1 : Y → Y {\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.
Examples Products of the underlying field Suppose that X {\displaystyle X} is a vector space over K {\displaystyle \mathbb {K} } , where K {\displaystyle \mathbb {K} } is either R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } and let T {\displaystyle T} be any set. Let Y := K T , {\displaystyle Y:=\mathbb {K} ^{T},} which is the product of K {\displaystyle \mathbb {K} } taken | T | {\displaystyle |T|} times, or equivalently, the set of all K {\displaystyle \mathbb {K} } -valued functions on T. Give Y {\displaystyle Y} its usual product topology, which makes it into a Hausdorff locally convex TVS. Then Y {\displaystyle Y} has the extension property. For any set T , {\displaystyle T,} the Lp space ℓ ∞ ( T ) {\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 – Function between topological vector spaces Hahn–Banach theorem – Theorem on extension of bounded linear functionals Hyperplane separation theorem – On the existence of hyperplanes separating disjoint convex sets
Citations
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