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Proving a set of derivatives to be a subset of real functions

  1. Aug 13, 2011 #1
    let C0 be the set of continuous functions f : R -> R. For n >= 1, let Cn denote theset of functions f : R -> R such that f is differentiable and such that f' is contained in C(n-1). (Therefore Cn is the set of functions whose derivatives f',f'',f''',....,f^(n) up to the nth order exist and are continuous.) Prove by induction that Cn is a subspace of V where V is the set of all functions f : R -> R.

    There are three properties that Cn must satisfy to be a subspace,
    1.) it must contain the zero vector of V
    2.) It must be closed under vector addition
    3.) it must be closed under scalar multiplication

    I am not sure which of these properties i must perform induction on (obviously not 1.) ) or should it be both 2.) and 3.)..?
    I would greatly appreciate it if someone could give me a hint for what the inductive step should be..?


  2. jcsd
  3. Aug 13, 2011 #2


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    Obviosuly Cn is a subset of V, so you just need to prove that it's a vector space by showing that the conditions for a vector space are satisfied. You don't need to do any induction.
  4. Aug 13, 2011 #3
    I am specifically asked to prove it by induction
  5. Aug 13, 2011 #4
    You'll need induction to show that the (f+g)^(n) = f^(n) + g^(n) and that (cf)^(n) = cf^(n) (which is what you need to prove to show that C^n is closed under addition and scalar multiplication). The inductive step would simply be using the linearity of derivation.
  6. Aug 13, 2011 #5


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    3 implies 1. And 2 and 3 can be combined into "closed under linear combinations".

    [tex](af+bg)^{(m+1)}=((af+bg)^{(m)})' =\dots[/tex]
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