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Math Help - Subspace of Infinitely Differentiable Functions on R

  1. #1
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    Subspace of Infinitely Differentiable Functions on R

    I have to prove which of the following are subspaces of infinitely differentiable functions on R:
    1) {f : f(0) = 0}
    2) {f : f'(0) = 0}
    3) {f : f(x) + f(-x) = 1}
    4) {f : f(x) + f(-x) = f(0)}

    I know that a subspace must be closed under addition and scalar multiplication, but I have no idea how to do this. How would you be able to test whether all of these are closed under addition and scalar multiplication?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    1) Denote E the vector space of infinitely differentiable functions on \mathbb{R} and F=\{f\in E:f(0)=0\} then:


    (a) The zero function 0:\mathbb{R}\to \mathbb{R},\;0(x)=0 for all x\in\mathbb{R} is infinitely differentiable on \mathbb{R} and satisfies 0(0)=0, so 0\in F .

    (b) For all f,g\in F the sum f+g is infinitely differentiable on \mathbb{R} and satisfies:

    (f+g)(0)=f(0)+g(0)=0+0=0

    so, f+g\in F.

    (c) For all \lambda\in \mathbb{R} and for all f\in F ...

    Show your work for the rest.


    Fernando Revilla
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