Prev: How to prove N+2k, k>1 Primes are infinite? #657 Correcting Math
Next: JSH: There is only one
From: Paul on 12 Jul 2010 19:43
I read that every separable metric space is homeomorphic to a subspace
of the Hilbert parallelotope and attempted to search online for a
proof of this.

The most relevant resource seemed to be http://www.math.jyu.fi/research/reports/rep90.ps
-- Geometric embeddings of metric spaces by Juha Heinonen.

Maybe, if I read this whole paper, the proof of the embedding theorem
would be apparent.

I see related theorems in this paper but nothing that leads directly
to the assertion.

This paper proves the existence of the Urysohn universal space, and
that every separable metric space embeds isometrically into
L_infinity.

Could anyone outline how I could find a proof of the assertion
regarding the Hilbert parallelotope, and whether the paper by Heinonen
is relevant, and if so how?

Thank you very much for your help.

Paul Epstein
From: spudnik on 12 Jul 2010 21:29
wow, a parallelotope associated with *every* Hilbert space?

> -- Geometric embeddings of metric spaces by Juha Heinonen.

thus&so:
yes, and prime gaps, sounds as if it could be related
to prime frequencies -- yahoo! (tm)
seriously, you're just talking about Venn diagrams; eh?
> There is only one; one is the loneliest number1

really, truly saith1

--BP's Next Bailout of Wall St. etc. ad vomitorium,
Waxman's *new* cap&trade (circa '91).
http://wlym.com
From: G. A. Edgar on 13 Jul 2010 07:35
In article
<f11d658b-4783-460a-bad0-5c4c8f0a89bf(a)s9g2000yqd.googlegroups.com>,
Paul <pepstein5(a)gmail.com> wrote:

> I read that every separable metric space is homeomorphic to a subspace
> of the Hilbert parallelotope and attempted to search online for a
> proof of this.
>
> The most relevant resource seemed to be
> http://www.math.jyu.fi/research/reports/rep90.ps
> -- Geometric embeddings of metric spaces by Juha Heinonen.

Probably a research paper assumes the reader already knows the basics
of the subject. Any text on beginning point-set topology will
have a proof of what you want. Look within or adjacent to material
on the Urysohn metrization theorem.

>
> Maybe, if I read this whole paper, the proof of the embedding theorem
> would be apparent.
>
> I see related theorems in this paper but nothing that leads directly
> to the assertion.
>
> This paper proves the existence of the Urysohn universal space, and
> that every separable metric space embeds isometrically into
> L_infinity.
>
> Could anyone outline how I could find a proof of the assertion
> regarding the Hilbert parallelotope, and whether the paper by Heinonen
> is relevant, and if so how?
>
> Thank you very much for your help.
>
> Paul Epstein

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
From: Paul on 13 Jul 2010 08:06
On Jul 13, 12:35 pm, "G. A. Edgar" <ed...(a)math.ohio-state.edu.invalid>
wrote:
> In article
> <f11d658b-4783-460a-bad0-5c4c8f0a8...(a)s9g2000yqd.googlegroups.com>,
>
> Paul <pepste...(a)gmail.com> wrote:
> > I read that every separable metric space is homeomorphic to a subspace
> > of the Hilbert parallelotope and attempted to search online for a
> > proof of this.
>
> > The most relevant resource seemed to be
> >http://www.math.jyu.fi/research/reports/rep90.ps
> > -- Geometric embeddings of metric spaces by Juha Heinonen.
>
> Probably a research paper assumes the reader already knows the basics
> of the subject.  Any text on beginning point-set topology will
> have a proof of what you want.  Look within or adjacent to material
> on the Urysohn metrization theorem.
>
>
>
>
>
>
>
> > Maybe, if I read this whole paper, the proof of the embedding theorem
> > would be apparent.
>
> > I see related theorems in this paper but nothing that leads directly
> > to the assertion.
>
> > This paper proves the existence of the Urysohn universal space, and
> > that every separable metric space embeds isometrically into
> > L_infinity.
>
> > Could anyone outline how I could find a proof of the assertion
> > regarding the Hilbert parallelotope, and whether the paper by Heinonen
> > is relevant, and if so how?
>
> > Thank you very much for your help.
>
> > Paul Epstein
>
> --
> G. A. Edgar                              http://www.math.ohio-state.edu/~edgar/- Hide quoted text -
>
> - Show quoted text -

Thanks, G.A. Edgar.

By the way, the link I gave is not to a research paper (despite the
title), and the prerequisites (beginning graduate) are less than you
might think.
[Not saying this to argue with you, but just that other non-expert
readers might want to look at Heinonen's paper.]

You're right that the proof of the Urysohn metrization theorem
contains a proof of the desired result at an intermediate stage. I
didn't know this before.

Paul Epstein
 | 
Pages: 1
Prev: How to prove N+2k, k>1 Primes are infinite? #657 Correcting Math
Next: JSH: There is only one