Statement:
Generalized Hahn-Banach Theorem: Assume that \(X\) is a real vector space, \((Z,\preccurlyeq)\) is a Dedekind complete ordered vector space and \(X_0\) is a subspace of \(X\). If \(\lambda_0 : X_0 \to Z\) is linear and \(p: X\to Z\) is sublinear and if \(\lambda_0 \preccurlyeq p\) on \(X_0\) then \(\lambda_0\) can be extended to a linear map \(\lambda : X\to Z\) such that \(\lambda \preccurlyeq p\) on \(X\). \ac{Schechter} \cite{1996b}
Howard_Rubin_Number: 372
Parameter(s): This form does not depend on parameters
This form's transferability is: Unknown
This form's negation transferability is: Negation Transferable
Article Citations:
Book references
Note connections:
Note 31
Definitions for forms [14 W], [70 A], [52 H] through [52 L], Form 372 and [372 A] through [372 D] .These are modifications of definitions from Schechter [1996a] and Schechter [1996b].
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 372 A | Existence of Banach Limits: For any directed set \((\Delta,\sqsubseteq)\) and any Dedekind complete ordered vector space \((Z,\preccurlyeq)\) there exists a \(Z\)-valued Banach limit. \ac{Schechter} \cite{1996b}. |
Note [31] |
| 372 B | Generalized Convex Domination Theorem: For every real vector space \(X\), every Dedekind complete ordered vector space \((Z,\preccurlyeq)\), every subspace \(X_0\subseteq X\), every linear \(\lambda_0 :X_0\to Z\) and every convex \(p: X\to Z\), if \(\lambda_0 \preccurlyeq p\) on \(X_0\) then \(\lambda_0\) can be extended to a linear map \(\lambda: X\to Z \) such that \(\lambda \preccurlyeq p\) on \(X\). \ac{Schechter} \cite{1996b}. |
Note [31] |
| 372 C | Generalized Support Theorem: For every real vector space \(X\) and every Dedekind complete ordered vector space \((Z,\preccurlyeq)\), every convex function \(p: X\to Z\) is a pointwise maximum of affine functions. \ac{Schechter} \cite{1996b}. |
Note [31] |
| 372 D | Generalized Sandwich Theorem: For every real vector space \(X\) and every Dedekind complete ordered vector space \((Z,\preccurlyeq)\), if \(e: X\to Z\) is a concave function, \(g: X\to Z\) is a convex function, and \(e\le g\) everywhere on \(X\), then there exists an affine function \(f: X \to Z\) satisfying \(e \le f \le g\) everywhere on \(X\). \ac{Schechter} \cite{1996b}. |
Note [31] |