Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Subspace question

  1. Jun 3, 2010 #1
    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 [tex]\in[/tex] F(S,F), such that f(s) = 0 for all but a finite number of elements in S (s [tex]\in[/tex] 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 :biggrin:.
  2. jcsd
  3. Jun 3, 2010 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Being 0 for all elements of S probably counts as being 0 for all but finitely many elements of S
  4. Jun 3, 2010 #3
    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?
  5. Jun 4, 2010 #4


    User Avatar
    Science Advisor

    Obviously not, since in that case you would not have a "0" vector.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook