Proving C(S,F) is a Subspace of F(S,F)

[Riemannliness]

Here's one I've been stewing over:
- Let S be a nonempty set of F, and F a field.
- Let F(S,F) be the set of all functions from S to the field F.
- Let C(S,F) denote the set of all functions f $$\in$$ F(S,F), such that f(s) = 0 for all but a finite number of elements in S (s $$\in$$ S).
Prove that C(S,F) is a subspace of F(S,F).

It's simple to show that the space is closed under addition and scalar multiplication, but I'm having a hard time finding a zero. It certainly isn't the zero function because that function is not nonzero at any finite number of points in S. I've played with a few ideas, but it always comes down to the ambiguity of the definition of S and the specifics of the finite nonzero points mapped by the functions. I can find functions that work for each specific case, but not one that works in all cases. I feel like I'm missing something relatively simple, so hints (BUT NOT ANSWERS) would be appreciated .

 

[Office_Shredder]

Being 0 for all elements of S probably counts as being 0 for all but finitely many elements of S

 

[Riemannliness]

You're probably right. But I was under the impression that it had to be nonzero at least a couple points. Would it be possible if that were the case?

 

[HallsofIvy]

Obviously not, since in that case you would not have a "0" vector.

 

[blue_raver22]

First of all, great job on attempting to prove the subspace property of C(S,F)! It is always important to question and analyze mathematical concepts, especially when they seem to be ambiguous or not straightforward. As you mentioned, it is easy to show that C(S,F) is closed under addition and scalar multiplication, so we just need to find a zero element to complete the proof.

To find the zero element, we need to consider the definition of C(S,F). It states that the functions in C(S,F) are only nonzero at a finite number of points in S. This means that for any function f in C(S,F), there exists a finite set of points in S where f is nonzero. Therefore, we can construct a zero element by considering a function that is zero at all points in S except for those finite points.

For example, let's say S = {1,2,3,4} and F = R (the set of real numbers). We can define a function f in C(S,F) as f(1) = 0, f(2) = 0, f(3) = 0, f(4) = 0. This function is only nonzero at a finite number of points in S (in this case, none) and therefore belongs to C(S,F). We can generalize this idea for any nonempty set S and field F.

In conclusion, the zero element for C(S,F) can be defined as the function that is zero at all points in S except for a finite number of points. This proves that C(S,F) is a subspace of F(S,F) since it satisfies all three properties of a subspace (closed under addition, scalar multiplication, and contains a zero element). Keep up the good work in questioning and analyzing mathematical concepts!