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).

Definitions

Throughout  and  will be topological vector spaces (TVSs) over the field  and  will denote the vector space of all continuous linear maps from  to , where if  and  are normed spaces then we endow  with its canonical operator norm.

Extensions

If  is a vector subspace of a TVS  then  has the extension property from  to  if every continuous linear map  has a continuous linear extension to all of . If  and  are normed spaces, then we say that  has the metric extension property from  to  if this continuous linear extension can be chosen to have norm equal to .

A TVS  has the extension property from all subspaces of  (to ) if for every vector subspace  of ,  has the extension property from  to . If  and  are normed spaces then  has the metric extension property from all subspace of  (to ) if for every vector subspace  of ,  has the metric extension property from  to .

A TVS  has the extension property if for every locally convex space  and every vector subspace  of ,  has the extension property from  to .

A Banach space  has the metric extension property if for every Banach space  and every vector subspace  of ,  has the metric extension property from  to .

1-extensions

If  is a vector subspace of normed space  over the field  then a normed space  has the immediate 1-extension property from  to  if for every , every continuous linear map  has a continuous linear extension  such that . We say that  has the immediate 1-extension property if  has the immediate 1-extension property from  to  for every Banach space  and every vector subspace  of .

Injective spaces

A locally convex topological vector space  is injective if for every locally convex space  containing  as a topological vector subspace, there exists a continuous projection from  onto .

A Banach space  is 1-injective or a -space if for every Banach space  containing  as a normed vector subspace (i.e. the norm of  is identical to the usual restriction to  of 's norm), there exists a continuous projection from  onto  having norm 1.

Properties

In order for a TVS  to have the extension property, it must be complete (since it must be possible to extend the identity map  from  to the completion  of ; that is, to the map ).

Existence

If  is a continuous linear map from a vector subspace  of  into a complete Hausdorff space  then there always exists a unique continuous linear extension of  from  to the closure of  in .
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  is an injective Banach space, then for every Banach space , every continuous linear operator from a vector subspace of  into  has a continuous linear extension to all of .

In 1953, Alexander Grothendieck showed that any Banach space with the extension property is either finite-dimensional or else  separable.

{{Math theorem|name=Theorem |math_statement=
Suppose that  is a Banach space over the field 
Then the following are equivalent:
  is 1-injective;
  has the metric extension property;
  has the immediate 1-extension property;
  has the center-radius property;
  has the weak intersection property;
  is 1-complemented in any Banach space into which it is norm embedded;
 Whenever  in norm-embedded into a Banach space  then identity map  can be extended to a continuous linear map of norm  to ;
  is linearly isometric to  for some compact, Hausdorff space, extremally disconnected space . (This space  is unique up to homeomorphism).

where if in addition,  is a vector space over the real numbers then we may add to this list:
  has the binary intersection property;
  is linearly isometric to a complete Archimedean ordered vector lattice with order unit and endowed with the order unit norm.
}}

Examples

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 .
Give  its usual product topology, which makes it into a Hausdorff locally convex TVS.
Then  has the extension property.

For any set  the Lp space  has both the extension property and the metric extension property.

See also

Citations

References

  
  
  
  

Topological vector spaces
Theorems in functional analysis