All subspaces closed ??

[don]

Just wondering if someone can point of the flaw in the following "proof" of : every subspace is closed.

Take X to be a real infinite dimensional Banach space. Let M be a subspace of X.

Let {x_i: i \in I } denote a Hamel basis for M and

{ x_i: i \in I U J } denote a Hamel basis for X.

Let P_i : X -> R denote the i^th coordinate projection.

Lets fist show that the kernel of P_i is closed. (I think this is where flas is).

For i isn't the kernel of P_i just a codimension 1 subspace and hence closed? (Just throw away the one direction x_i)

Then M = int_{i \in J } ker(P_i)

( the intersection of the kernel of P_i where i ranges over J )

and so M closed.

any comments would be greatly appreciated

craig

[Anton Deitmar]

Why should a codimension one subspace be closed?

Anton

[G. A. Edgar]

In article
<25994205.1155188563...@nitrogen.mathforum.org>, don
<ctc...@sfu.ca> wrote:

> Just wondering if someone can point of the flaw in the following "proof" of :
> every subspace is closed.
>
>
>
> Take X to be a real infinite dimensional Banach space. Let M be a subspace
> of X.
>
> Let {x_i: i \in I } denote a Hamel basis for M and
>
> { x_i: i \in I U J } denote a Hamel basis for X.
>
>
> Let P_i : X -> R denote the i^th coordinate projection.
>
> Lets fist show that the kernel of P_i is closed. (I think this is where flas
> is).

correct, this is the flaw

>
> For i isn't the kernel of P_i just a codimension 1 subspace and hence
> closed? (Just throw away the one direction x_i)

in general such a subspace need not be closed

>
>
> Then M = int_{i \in J } ker(P_i)
>
> ( the intersection of the kernel of P_i where i ranges over J )
>
> and so M closed.
>
>
>
>
> any comments would be greatly appreciated
>
> craig

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/

[David C. Ullrich]

On Thu, 10 Aug 2006 01:42:12 EDT, don <ctc...@sfu.ca> wrote:

>Just wondering if someone can point of the flaw in the following "proof" of : every subspace is closed.
>
>
>
>Take X to be a real infinite dimensional Banach space. Let M be a subspace of X.
>
>Let {x_i: i \in I } denote a Hamel basis for M and
>
>{ x_i: i \in I U J } denote a Hamel basis for X.
>
>
>Let P_i : X -> R denote the i^th coordinate projection.
>
>Lets fist show that the kernel of P_i is closed. (I think this is where flas is).
>
>For i isn't the kernel of P_i just a codimension 1 subspace and hence closed? (Just throw away the one direction x_i)

Can you write down an actual _proof_ that a codimension-1 subspace is
closed?

(Hint: You can't. There exist discontinuous linear functionals...)

>
>Then M = int_{i \in J } ker(P_i)
>
>( the intersection of the kernel of P_i where i ranges over J )
>
>and so M closed.
>
>
>
>
>any comments would be greatly appreciated
>
>craig

************************

David C. Ullrich

[Dave L. Renfro]

don wrote (in part):

> For i isn't the kernel of P_i just a codimension 1 subspace
> and hence closed? (Just throw away the one direction x_i)

There exist non-Borel and dense codimension 1 subspaces
of (any, I believe) infinite-dimensional Banach spaces.

There's a paper by Victor Klee, which I thought was fairly
well known (but I only got 5 google hits for its title),
that gives quite a number of exotic possibilities for
proper subspaces of infinite dimensional Banach spaces:

Victor L. Klee, "Dense convex sets", Duke Mathematical Journal
16 #2 (June 1949), 351-354.

This was followed by two or three other papers (at least),
but I don't remember enough about them to find them.
[Billy J. Pettis? I have the papers at home in a folder
of results related to Klee's paper.] I suspect that anyone
who has access to Mathematical Reviews can easily come
up with the references, but this also isn't available
to me where I'm at.

I think one of the results Klee (or maybe someone following
him) proved is that you can always find infinitely many
[uncountably many? continuum many?] pairwise disjoint
non-Borel and dense codimension 1 subspaces in an
infinite-dimensional Banach space. I'm sure there are
others in here who know a lot more about this than I
do, as I'm not very knowledgeable in functional analysis.

Dave L. Renfro

[Robert Israel]

In article <1155220951.2...@m73g2000cwd.googlegroups.com>,

It can't be that.
You can't even find two disjoint linear subspaces of any
vector space: they all contain 0. And the intersection of
two distinct codimension 1 subspaces is a codimension 2
subspace.

Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

[Dave L. Renfro]

Dave L. Renfro wrote:

>> I think one of the results Klee (or maybe someone following
>> him) proved is that you can always find infinitely many
>> [uncountably many? continuum many?] pairwise disjoint
>> non-Borel and dense codimension 1 subspaces in an
>> infinite-dimensional Banach space. I'm sure there are
>> others in here who know a lot more about this than I
>> do, as I'm not very knowledgeable in functional analysis.

Robert Israel wrote:

> It can't be that.
> You can't even find two disjoint linear subspaces of any
> vector space: they all contain 0. And the intersection of
> two distinct codimension 1 subspaces is a codimension 2
> subspace.

Ooops! Well, I guess Klee's results were for the more
inclusive class of convex subsets, seeing as how that
was the title of his paper. Nonetheless, there do exist
non-Borel (that are also dense in the space, I think)
subspaces of infinite-dimensional Banach spaces
(which I'm sure you know -- this was so that the
original poster doesn't think the extreme failure
of "closed" I mentioned might be in doubt).

Dave L. Renfro