Re: A Pythagorean Question

From: Ted Hwa (hwatheod_at_xenon.Stanford.EDU)
Date: 03/18/05 

Date: Fri, 18 Mar 2005 04:37:17 +0000 (UTC)

Edward Green <spamspamspam3@netzero.com> wrote:

: It is a common thing to do to extend this result to n-dimensional and
: even countably infinite dimensional vector spaces. My question: is
: there a natural definition of an appropriate vector norm in higher
: dimensions which does _not_ simply reiterate the requirement to take
: the sum of squares, so making "|x|^2 = (x_1)^2 + (x_2)^2 + ... +
: (x_i)^2 + ... " a theorem, or is it better regarded as simply a
: plausible definition? Or somewhere in between?

As I see it, the question is how you define "n-dimensional space" in
the first place. If you define it as the set of n-tuples of real
numbers, then I think you may be stuck with the Pythagorean theorem
being the definition of distance. After all, "distance" is what
makes R^n into Euclidean space and not just a bunch of n-tuples :))
It's also what makes a normed space a normed space and not just a
vector space, and so on. I don't think you can get around this.

If you want to prove it as a theorem, you'd have to do something
such as define n-dimensional space with axioms a la Euclid,
axiomatizing distance, congruence, and so on, and then you could
prove the Pythagorean theorem as a bonafide theorem.

Ted

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