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- 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). (en)
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- 8756 (xsd:nonNegativeInteger)
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- 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 . .
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. (en)
- Suppose that is a Banach space with the metric extension property.
Then the following are equivalent:
# is reflexive;
# is separable;
# is finite-dimensional;
# is linearly isometric to for some discrete finite space (en)
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- 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). (en)
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- Vector-valued Hahn–Banach theorems (en)
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