From: a_steiner@my-deja.com
Subject: Re: Weierstrass Theroem
Date: Thu, 31 Aug 2000 18:55:36 GMT
Newsgroups: sci.math
Summary: [missing]

In article <8oh2jk$clj@mcmail.cis.McMaster.CA>,
  kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) wrote:
> In article <8oh018$sss$1@nnrp1.deja.com>,  <liliano@my-deja.com> wrote:
> :Hi,
> :I don't understand the demostration of Weierstrass Theroem.
> :(max and min existence)
> :Can you help me ?
> :
> :Thank you
> :Liliano
>
> I'd like to help but my problem is that I don't know what the wording and
> the context was (many theorems are attributed to Weierstrass, and many
> variants). Also, what is the particular proof that you did not understand,
> and what are the previous theorems that the proof is quoting?
>
> Is it the one that says
>
>  "every continuous function on a closed bounded interval of real numbers
> attains its maximal and its minimal value"
>
> or
>
>  "every continuous function on a compact metric space attains its max. and
> min."
>
> or the same with "compact topological space",

I've always uknown these theorems as Bolzano Intermediate Value
Theorems. I didn't know they were also attributed to Weiestrass. At
least in Real Analysis.

Two important theorems attributed two Wierstrass (and also Bolzano) I
have studied (also in Analysis) are

1)Every bounded set in a real vector space has an accumulation (or
cluster) point. I think this theorem also holds for arbitrary
topological spaces

2) Every bounded sequence in a real vector space has a convergent sub-
sequence. I think this theorem is valid in any metric space but I'm not
sure if it is true in any arbitrary toplogical space. I guess it is
always true if any convergent sequence in the topological has oly one
limit (I'm not sure now, but I think those spaces are called Hausdorff
spaces

At least in Analysis, the second theorem is just a consequence, almost
a corollary, of the first.

Artur


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From: hook@nas.nasa.gov (Ed Hook)
Subject: Re: Weierstrass Theroem
Date: 31 Aug 2000 21:00:42 GMT
Newsgroups: sci.math

In article <8om9ms$5j5$1@nnrp1.deja.com>, a_steiner@my-deja.com writes:
|> 
|> I've always uknown these theorems as Bolzano Intermediate Value
|> Theorems. I didn't know they were also attributed to Weiestrass. At
|> least in Real Analysis.

|> Two important theorems attributed two Wierstrass (and also Bolzano) I
|> have studied (also in Analysis) are

|> 1)Every bounded set in a real vector space has an accumulation (or
|> cluster) point. I think this theorem also holds for arbitrary
|> topological spaces

  No. For starters, the statement
  only makes sense in metric spaces
  (or, at least, spaces where you can
  talk about "boundedness").


|> 2) Every bounded sequence in a real vector space has a convergent sub-
|> sequence. I think this theorem is valid in any metric space but I'm not
|> sure if it is true in any arbitrary toplogical space. I guess it is
|> always true if any convergent sequence in the topological has oly one
|> limit (I'm not sure now, but I think those spaces are called Hausdorff
|> spaces

   This does _not_ hold in every
   metric space (and so it's certainly
   not true in an arbitrary topological
   space). Your comment about Hausdorff
   spaces is correct, though ...

   The theorem that you _might_ be
   thinking of:

   Thm. The following are equivalent for
   a topological space X:

   (1) X is _countably_compact_

   (2) Every countably infinite subset
       of X has an accumulation point

   (3) Every sequence in X has an
       accumulation point.

   Here, X is countably compact if every
   _countable_ open cover of X has a 
   finite subcover.

   If X happens to be a metric space,
   then X is countably compact iff X
   is compact. And, in the case of a
   metric space (at least), you can
   extract a subsequence converging to
   any accumulation point of a given
   sequence ...


|> At least in Analysis, the second theorem is just a consequence, almost
|> a corollary, of the first.

|> Artur


-- 
 Ed Hook                              |       Copula eam, se non posit
 Computer Sciences Corporation        |         acceptera jocularum.
 NAS, NASA Ames Research Center       |    All opinions herein expressed are
 Internet: hook@nas.nasa.gov          |              mine alone