Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    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

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

  2. jcsd
  3. Oct 14, 2011 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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


    User Avatar
    Science Advisor

    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


    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook