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Topic: Every Subspace of R^N has an Orthogonal Basis?
Replies: 175   Last Post: Jul 4, 2006 9:20 AM

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Denis Feldmann

Posts: 533
Registered: 4/23/06
Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal
Basis?)

Posted: Jul 2, 2006 4:39 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Herman Rubin a écrit :
> In article <44a7eabb$1$fuzhry+tra$mr2ice@news.patriot.net>,
> Shmuel (Seymour J.) Metz <spamtrap@library.lspace.org.invalid> wrote:

>> In <e80t3h$ai5$1@panix2.panix.com>, on 06/29/2006
>> at 11:54 AM, lrudolph@panix.com (Lee Rudolph) said:

>
>>> I am using "the set X has a cardinality" to mean "there is a
>>> bijection between X and a cardinal number", where a "cardinal number"
>>> is an initial ordinal.

>
>> While I don't like that definition, I agree that it makes your
>> statement meaningful.

>
> It does, but it makes a lot of things more complicated.
> With it, one cannot discuss cardinalities of sets which
> cannot be well-ordered.
>
> There is the rank definition of Scott if the sets of
> every given rank form a set.
>
> But the ideas of comparing the cardinalities of sets do
> not depend on any of these, and are what are used in practice.
>

>>> You appear to be using "cardinality" to mean "equivalence class
>>> under bijection". Qualms (perhaps unjustified) about proper classes
>>> keep me from talking that way these days.

>
>> I understand the qualms, but to me it appears to be throwing out the
>> baby with the bath water.

>
> Agreed. The baby does not need a name.
>

>> IAC, I'm hoping that someone has a definitive answer for Aleph_0 <
>> dimension < C.

>
> Even with the Axiom of Choice, this cannot be answered.


This? With choice, it is pretty obvious that any infinite set can be
made into a Q-vector space, which was the original question, IIRC...


> There are some alephs which C cannot be, but "most" of
> them work. Also, aleph_1 and C might be incomparable,
> but we can say that aleph_1 < 2^(2^C).



Date Subject Author
6/26/06
Read Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Lee Rudolph
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Jose Carlos Santos
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Jose Carlos Santos
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Robert Low
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
James Dolan
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Gene Ward Smith
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Brian Quincy Hutchings
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Michael L. Siemon
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Michael L. Siemon
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Tim Golden http://bandtech.com
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Tim Golden http://bandtech.com
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Tim Golden http://bandtech.com
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Robert Low
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Tim Golden http://bandtech.com
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Dave Seaman
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Pinky
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Gene Ward Smith
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Gene Ward Smith
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Gene Ward Smith
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Gene Ward Smith
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Gene Ward Smith
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Mariano
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/29/06
Read "dimensionality" v. "dimension" (was, Re: Every Subspace of R^N has an Orthogonal Basis?)
Lee Rudolph
6/29/06
Read Re: "dimensionality" v. "dimension" (was, Re: Every Subspace of R^N has an Orthogonal Basis?)
K. E. Pledger
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Denis Feldmann
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Denis Feldmann
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Gene Ward Smith
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Jose Carlos Santos
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Robert Low
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Robert Low
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/26/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
kunzmilan@atlas.cz
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Robert Low
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Lee Rudolph
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Denis Feldmann
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Robert Low
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/30/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/30/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Denis Feldmann
6/30/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/30/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Denis Feldmann
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Robert Low
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Denis Feldmann
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/27/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Robert Low
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Robert Low
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Gene Ward Smith
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Lee Rudolph
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Mariano
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Mariano
6/29/06
Read Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal Basis?)
Lee Rudolph
6/29/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal Basis?)
magidin@math.berkeley.edu
6/29/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal Basis?)
Shmuel (Seymour J.) Metz
6/29/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal Basis?)
Lee Rudolph
6/30/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal Basis?)
Bertie Reed
6/30/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal
Basis?)
Denis Feldmann
6/30/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal Basis?)
Mariano
6/30/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal
Basis?)
Denis Feldmann
6/30/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal Basis?)
Mariano
6/30/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal
Basis?)
Aatu Koskensilta
7/2/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal Basis?)
Shmuel (Seymour J.) Metz
7/2/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal Basis?)
Herman Rubin
7/2/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal
Basis?)
Denis Feldmann
6/29/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal Basis?)
Mariano
6/29/06
Read Re: Cardinals of fields (was, Re: Every Subspace of R^N has an Orthogonal Basis?)
Gene Ward Smith
6/28/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Robert Low
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Lee Rudolph
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Lee Rudolph
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
W. Dale Hall
6/30/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Robert Low
6/30/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/30/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/30/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/30/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/30/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
7/1/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
7/1/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
7/2/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
7/2/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Denis Feldmann
7/2/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
7/2/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
7/2/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
7/3/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Gerry Myerson
7/3/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
7/3/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
7/3/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
7/3/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
7/3/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
magidin@math.berkeley.edu
7/3/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
7/3/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
7/3/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
David C. Ullrich
6/30/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Robert Low
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Hatto von Aquitanien
6/29/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Virgil
7/4/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
kunzmilan@atlas.cz
7/4/06
Read Re: Every Subspace of R^N has an Orthogonal Basis?
Lee Rudolph

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