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Vector spaces of functions

  1. Oct 14, 2011 #1
    1. The problem statement, all variables and given/known data

    V is the set of functions R -> R; pointwise addition and (a.f)(x) = f(ax) for all x.

    is V a vector space given the operations?


    2. Relevant equations

    nil.

    3. The attempt at a solution

    i think it is not closed under multiplication.
    if r is an element of R, then
    r*a(x) . r*f(x) = (r^2)*(a.f)(x)
    which is not equal to
    r*f(ax)

    im not really sure if i even have the correct approach.
    any help would be greatly appreciated.

    thanks!
     
  2. jcsd
  3. Oct 14, 2011 #2

    LCKurtz

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    There are lots of relevant equations -- they comprise the definition of a vector space.
    I don't follow your argument. There is no "multiplication" of vectors in the definition of a vector space, only addition. All you need to do is pick one of the properties of a vector space that doesn't work and give a counter-example. Which property are you working with above? It might be useful to list them.
     
  4. Oct 14, 2011 #3

    Deveno

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    suppose that f(x) = x2.

    is it the case that ((a+b).f)(x) = (a.f)(x) + (b.f)(x)?

    (this is the distributivity of field addition over scalar mutliplication axiom).
     
  5. Oct 14, 2011 #4

    Mark44

    Staff: Mentor

    If I'm understanding the problem correctly, f(x) = x2 is not a member of set V, since af(x) [itex]\neq[/itex] f(ax).
     
  6. Oct 15, 2011 #5

    LCKurtz

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    You aren't. The vector space is the set of [all] functions from R to R. It's just that scalar multiplication is defined in an unusual way.
     
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