Re: Please i need the The Separable Hahn-Banach Theorem proof
- From: Ronald Bruck <bruck@xxxxxxxxxxxx>
- Date: Sat, 29 Apr 2006 23:47:28 -0700
In article <1146329955.401025.320710@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Ziko <solmaths@xxxxxxxxxxx> wrote:
Hi
The Separable Hahn-Banach Theorem: if f is a bounded linear
functional
on subspace Z of normed space H then f has an extension
f^ to the whole space with the same norm.
is there a proof ..? ( without using Zorn's Lemma).
Probably not without Zorn's Lemma (or other form of AC). Perhaps
someone will weigh in on whether it's EQUIVALENT to AC?
I do remember a development by Alberto Calderon in which he proved
separation for two-dimensional spaces, then (with Zorn's Lemma) showed
how that implied the general form.
I don't know why you call it the Separable HB Theorem. You didn't
mention separability. If you HAVE separability then of course you can
do it without Zorn's Lemma, because a strictly ascending chain of
subspaces has to be countable. (But probably have to use countable AC.
I think you have to use THAT to breathe!)
--
Ron Bruck
.
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