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Subspaces of functions

  1. Sep 13, 2005 #1
    Yet another problem I need to get some starting help on:

    Show that the set of continuous functions f=f(x) on [a,b] such that [tex]\int \limits_a^b f(x) dx=0[/tex] is a subspace of C[a,b]
    Thank you
     
  2. jcsd
  3. Sep 13, 2005 #2

    NateTG

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    I would start by checking the definition of subspace.
     
  4. Sep 13, 2005 #3
    Definition of subspace means that the functions are closed under addition and scalar multiplication
     
  5. Sep 13, 2005 #4

    NateTG

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    So can you show that that's true for the potential subspace in your example?
     
  6. Sep 13, 2005 #5
    You mean that if [tex]\int \limits_a^b f(x) dx=0[/tex] and [tex]\int \limits_a^b g(x) dx=0[/tex], can I prove that [tex]\int \limits_a^b f(x)+g(x) dx=0[/tex]? Also, if [tex]\int \limits_a^b f(x) dx=0[/tex] then [tex]k\int \limits_a^b f(x) dx=0[/tex]?
     
  7. Sep 13, 2005 #6

    NateTG

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    Yeah, that's pretty much it. (Technically you also have to show that it's a subset, but in this case that's trivial.)
     
  8. Sep 13, 2005 #7
    Thank you!

    The "hard ones" are so easy sometimes.....
     
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