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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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Hatto von Aquitanien

Posts: 582
Registered: 11/26/05
Re: Every Subspace of R^N has an Orthogonal Basis?
Posted: Jun 27, 2006 4:53 AM
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Gene Ward Smith wrote:

>
> Hatto von Aquitanien wrote:

>> Gene Ward Smith wrote:
>
>> > Consider all the continuous real-valued functions on the interval [0,
>> > 1]. If we define f+g by
>> > (f+g)(x) = f(x)+g(x), and k*f by (k*f)(x) = k*f(x), these satisfy the
>> > axioms for a real vector space.

>>
>> In R^n? n is finite in this discussion. It may not have been stated
>> explicitly, but it should be clear from the context upon review.

>
> What rubbish; there was no such implication.


Curious. Perhaps this is another one of those cultural differences. I have
always taken {1,2,...,n} to designate a finite range of positive integers,
and therefore, by implication R^n to be a set of n-tuples with a finite
number of components. In order for me to consider the possibility that n
tends to infinity, it would have to be stated explicitly, or be obvious
from the context. So, let me be clear. Edwards was fairly specific about
this being a finite dimensional space.

> However, we can do
> something similar for finite dimensions. Consider the real vector space
> of all real-analytic functions on R. Take the kernel of the derivative
> operator applied thrice: that is, take all the real-analytic functions
> f for which D^3(f) is the zero function. There's your finite
> dimensional space.
>
> If you don't like that one, how about this: all functions of the form A
> sin(x + B) for A and B any real number.


What is x?
--
Nil conscire sibi


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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