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Determine if L is a vector space

  1. May 12, 2016 #1
    1. The problem statement, all variables and given/known data
    V = function space from R to R
    L ={ f in V | f(1/2) > f(2) }
    Determine if L is a vector space.


    2. Relevant equations


    3. The attempt at a solution
    1. Closed under addition: Do i do addition like this let g and e in V, then g(1/2)+e(1/2) > g(2) + e(2) but the addiction of two functions is even a function ??
     
  2. jcsd
  3. May 12, 2016 #2

    Ray Vickson

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    Of course it is. After all, what is a function, anyway?

    Besides addition, you also need to look at multiplication by a scalar, that is, you need to look at the function c*f(x), where c is a number and f is a function.
     
  4. May 12, 2016 #3
    So as the way i have shown above is enough for proving its closed under addition?
    let a be a scaler, and f be in V then a*(f(1/2)>a*f(2) is that enough for showing it's closed under scalar multiplication ? looks too much trivial
     
  5. May 13, 2016 #4

    vela

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    If your claim were true, yes, that's all you have to do. You should think about the different types of values ##a## can assume.
     
  6. May 13, 2016 #5
    If a = 0 inequality won't hold
    Therefore it's not vector space above r?
     
  7. May 13, 2016 #6

    Ray Vickson

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    If f(1/2) = 10 and f(2) = 2, we have f(1/2) > f(2) because 10 > 2. Do we also have -10 > -2?
     
  8. May 13, 2016 #7

    vela

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    Yes, that's right. You can also use what Ray pointed out.
     
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