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 or the complex numbers ) to linear operators valued in topological vector spaces (TVSs).
Throughout X and Y will be topological vector spaces (TVSs) over the field 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.
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 [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 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 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.
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 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.
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 from Y to the completion Z of Y; that is, to the map Z → Y). [1]
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. [1] [2] Consequently, it suffices to only consider maps from closed vector subspaces into complete Hausdorff spaces. [1]
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 Then the following are equivalent:
where if in addition, Y is a vector space over the real numbers then we may add to this list:
Theorem [1] — Suppose that Y is a real Banach space with the metric extension property. Then the following are equivalent:
Products of the underlying field
Suppose that is a vector space over , where is either or and let be any set. Let which is the product of taken times, or equivalently, the set of all -valued functions on T. Give its usual product topology, which makes it into a Hausdorff locally convex TVS. Then has the extension property. [1]
For any set the Lp space has both the extension property and the metric extension property.
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 or the complex numbers ) to linear operators valued in topological vector spaces (TVSs).
Throughout X and Y will be topological vector spaces (TVSs) over the field 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.
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 [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 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 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.
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 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.
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 from Y to the completion Z of Y; that is, to the map Z → Y). [1]
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. [1] [2] Consequently, it suffices to only consider maps from closed vector subspaces into complete Hausdorff spaces. [1]
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 Then the following are equivalent:
where if in addition, Y is a vector space over the real numbers then we may add to this list:
Theorem [1] — Suppose that Y is a real Banach space with the metric extension property. Then the following are equivalent:
Products of the underlying field
Suppose that is a vector space over , where is either or and let be any set. Let which is the product of taken times, or equivalently, the set of all -valued functions on T. Give its usual product topology, which makes it into a Hausdorff locally convex TVS. Then has the extension property. [1]
For any set the Lp space has both the extension property and the metric extension property.