Is \Re^\infty Used for Sets with Zero Elements?

[Palindrom]

Is it correct that $$\[ \Re ^\infty = \left\{ {\xi \in \Re ^N \left| {\exists n \in N:\forall k > n\,\xi _k = 0} \right.} \right\} \]$$, N denoting the natural numbers?
If not, is there another symbol for the left hand side of the equation?
Is this also used with any other set containing a zero element?

 

[matt grime]

try writing it more clearly in english and not latex so that it gets the spacing correct. As it is you appear to be asking for the correct notation for a countable array (x,y,z,.) where all but finitely many terms are zero. This is the coproduct you want, not the product.

$$\oplus \mathbb{R}^i \ i \in \mathbb{N}$$

 

[Palindrom]

That's exactly what I wanted. Could you please define coproduct?
Oh, and what was the problem with the spacing?

 

[matt grime]

it looked like "t times zeta" you see.

the coproduct is exactly "coutable array" with all but finitely many entries zero. it has a better categorical universality definition. given sets X_i i in I some index the coporduct, when it exists is the object sum X_i with maps from the X_i to the sum X_i that are "inclusions" and that is universal with respect to this definition

sounds nasty, right?

eaiser to think in terms of finite (co)products of vector spaces.

if X and Y are vector spaces, then the coproduct is a vector space with inclusions from X and Y that is smallest with this relation and no other. Thus it is precisely the set of ordered pairs (x,y) with x in X and y in Y, ie if X=Y=R then it is R^2., and the inclusions are x to (x,0) and y to (0,y)

The product is the dual defintion, that is it comes with projections onto the factors X and Y, and it so happens that for a finite number of spaces this is the same as the coproduct, and the maps are


(x,y) to x and (x,y) to y.

when you pass to infinitely many factors they are different.

the product over i of in N R_i is the set of all arrays (x_1,x_2,x_3,...) with x_i in R_i


the coproduct is the set of arrays (x_1,x_2,...) where only finitely many are nonzero (or equivalently x_i is zero for all i sufficiently large).

 

[Palindrom]

O.K., great, that's just what I wanted to know, thanks.

One last question on the subject- what happens if X_i don't have a zero element?

 

[matt grime]

We are talking about these things as *vector spaces* and not as sets so they must have the zero vector in them. It was as a vector space that i took the coproduct, since *you* wanted to the zero element to be distinguished in this way. I wansn't talking about the set coproduct. (see below)


However, in general, the coproduct depends upon the category in which you are working; part of the definition of coproduct is the existence of certain maps, and the form of these maps is determined by the category.

In the category SET the coproduct is called the disjoint union. and is not the same as the coproduct of vector spaces.

 

[Palindrom]

Thank you so much, you've been of great help.