20061009, 18:43  #1 
Cranksta Rap Ayatollah
Jul 2003
641 Posts 
S^oo ?
I have a homework problem regarding and while I'm not looking for anyone to do my homework, I don't know how to define . The only two definitions I know for are the points that are distance 1 from the origin in and the boundary of , neither of which are helping me for the infinite case. Any info would be greatly appreciated
P.S. Wow, that's an attractive avatar. 
20080304, 16:14  #2 
Feb 2007
432_{10} Posts 
(to reactivate this subforum without news since beg. of December...)
If you suggest the context of S^n in IR^n, I suppose S^oo is IR^IN, (vectors with oo number of components, a.k.a. sequences), probably restricted to a subspace with finite norm. But since not all norms are equivalent in oodimensional case, one must know which one is meant. It could be the \ell^2 norm (sqrt of the infinite sum of squared components which is required to converge) or the max norm or some ultrametric norm like limsup x_n^{w_n} where (w_n) can be any sequence of weights (e.g. w_n=1/log(n) gives sequences of at most polynomial growth, and that ultranorm gives (exp() of) the corresponding power of n). Last fiddled with by m_f_h on 20080304 at 16:40 
20080304, 16:40  #3 
(loop (#_fork))
Feb 2006
Cambridge, England
6,323 Posts 
I'd have thought S^\infty was the set of all vectors of real numbers the sums of whose squares is 1 ... what is it you need to prove about it?

20080304, 17:45  #4 
Feb 2007
2^{4}·3^{3} Posts 
you mean: the set of all vectors of any length? well vectors of finite length can always be considered as sequences with only a finite number of nonzero components. so, it is essentially what I said. (with \ell^2 norm, where the sum of squared components of course converges if there is only a finite number of nonzero ones.).

20080305, 00:03  #5 
Cranksta Rap Ayatollah
Jul 2003
1201_{8} Posts 
Thanks for the timely response.
So, if we take and two line segments, we can identify the boundaries of the line segments with and obtain . Similarly, if we take and two discs, we can identify the boundaries of the two discs with and obtain . We can carry this process on indefinitely, and define to be what we get in the limit Last fiddled with by Orgasmic Troll on 20080305 at 00:04 
20080308, 20:20  #6  
Feb 2007
1B0_{16} Posts 
Quote:
Also, it does not really give me an intuitive idea of how the limit would look like... Finally, what is the definition of "limit" used here ? Last fiddled with by m_f_h on 20080308 at 20:22 

20080309, 01:40  #7  
Cranksta Rap Ayatollah
Jul 2003
641 Posts 
Quote:


20080313, 21:18  #8 
Feb 2007
2^{4}·3^{3} Posts 
This notation does not make sense. You know that things like "the set of all sets" lead to logical inconsistency (if it is a set, it cannot be included in itself). You have to specify in which "universe" the objects you are considering live in.
Also, even assuming that we have given a rigorous meaning to this notation, it would imply that any element of S°° would be the element of some S^n, and reciprocally. This also seems not obvious / correct to me. (Why should a 1sphere also be a oosphere ? It is not a 2sphere, though.) 
20080314, 06:03  #9  
Cranksta Rap Ayatollah
Jul 2003
641 Posts 
Quote:
Sure, every element of is an element of (an infinite number of) . Every element of the reals is also contained in an infinite number of intervals of the form [a,a]. I don't know where you're getting the notion that a 1sphere is a sphere. pi is a real number, but that doesn't make it all of the reals. 

20080314, 14:07  #10 
"William"
May 2003
New Haven
2^{2}·3^{2}·5·13 Posts 
The sequence x_{n} = 2^{n/2} has the sum of squares = 1 but is not in any of the S^{n}  although it IS in the completion of the Union.

20080329, 17:25  #11 
Aug 2002
Ann Arbor, MI
433 Posts 
You have two main choices: One is just to take the union of all the S^n's (basically infinite sequences that are zero for all but finitely many terms, where the sum of the squares of the nonzero terms is 1), or in a sense taking the completion of this, and allowing all infinite sequences where the sum of the squares of the entries add up to 1 (allowing things like 6/pi^2*(1,1/2,1/3,1/4,...) .
If you know topology, it's basically the difference between choosing the product topology and the box topology (http://mathworld.wolfram.com/ProductTopology.html). I have no idea which way is the standard way to define S^inf, and without knowing the question we can't really judge which one makes more sense in this situation. 