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Form equivalence class Howard-Rubin Number: 0

Statement:

Assume \(V\) is a separable vector space, \(D\) is a countable dense subset of \(V\), and \(p :V \to \Bbb R\) satisfies\(p(x+y) \le p(x) + p(y)\), and\((\forall t\ge 0)(\forall x\in V)(p(tx) = tp(x))\). Also assume that \(p\) satisfies the following condition:

\((\forall \epsilon > 0)(\forall x\in V)(\exists y\in D)\) such that: \(p(x - y) < \epsilon \mathrm{\; and \; } p(y - x) < \epsilon.\) Moreover, if \(f\) is a linear function from a subspace \(S\) of \(V\) into \(\Bbb R\) which satisfies \((\forall x \in S)(f(x) \le p(x))\), then \(f\) can be extended to \(f^* :V \to \Bbb R\) so that \(f^* \) is linear and\((\forall x \in V)(f^*(x) \le p(x))\).

Howard-Rubin number: 0 AM

Citations (articles):

Connections (notes): Note [143] Form [0 AM], a form of the Hahn-Banach Theorem for separable vector spaces, is proved to be provable in ZF\(^{0}\)

References (books):

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