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Subsets and subspace

  1. Mar 3, 2009 #1
    1. The problem statement, all variables and given/known data

    Which of the following subsets of the vector space R^R of all functions from R to R are subspaces? (proofs or counterexamples required)

    U:= f R^R, f is differentiable and f'(0) = 0

    V:= fR^R, f is polynomial of the form f=at^2 for some aR
    = There exists a of the set R: for all s of R: f(s) = as^2

    W:= " " f is polynomial of the form f=at^i for some aof the set R and i of the set N
    = there exists i of N, there exists a of R: that for all s of R: f(s) = as^i

    X:= " " f is odd
    (f is odd such that f(-s) =-f(s) for all s of R

    2. Relevant equations

    3. The attempt at a solution
    U: i am not sure about 0 , is it differentiable if f=0
    V,W: is that 0 belongs to the polynomial, in other words i am not sure about the definition of polynomial
    X: 0 is an odd function ? I knew that odd+odd=odd function a*odd=odd function
  2. jcsd
  3. Mar 3, 2009 #2


    Staff: Mentor

    U -- f(x) = 0 is differentiable, and its derivative is 0.
    V, W -- a zero polynomial, such as 0 + 0t + 0t^2, is perfectly valid.
    X - The 0 function is both even and odd (since f(-x) = f(x), it's even, and since f(-x) = -f(x), it's odd).

    By the way, your vector space is R X R, I believe, not R^R
  4. Mar 3, 2009 #3
    oh i c, thx a lot
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