Implications

We have the following indirect implication of form equivalence classes:

287 \(\Rightarrow\) 287
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
287 \(\Rightarrow\) 287

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
287:

The Hahn-Banach Theorem for Separable Normed Linear Spaces:  Assume \(V\) is a separable normed linear space and \(p :V \to \Bbb R\) satisfies \(p(x+y) \le p(x) + p(y)\) and \((\forall t \ge 0)(\forall x \in V)(p(tx) = tp(x))\) and assume \(f\) 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 \((\forall x \in V)(f^*(x) \le p(x))\).

287:

The Hahn-Banach Theorem for Separable Normed Linear Spaces:  Assume \(V\) is a separable normed linear space and \(p :V \to \Bbb R\) satisfies \(p(x+y) \le p(x) + p(y)\) and \((\forall t \ge 0)(\forall x \in V)(p(tx) = tp(x))\) and assume \(f\) 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 \((\forall x \in V)(f^*(x) \le p(x))\).

Comment:

Back