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Linear Functionals

  1. Nov 20, 2007 #1
    How do I solve this problem- I know it has something to do Riesz represenation but am having difficulty connecting dots

    Conside R4 with usual inner product. Find the linear funcitonal associated to the vector (1,1,2,2).

    What am I missing- is this problem complete or is there something else Also what does usual inner product mean.

    Sharkie
     
  2. jcsd
  3. Nov 21, 2007 #2

    morphism

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    Did you read the statement of the Riesz representation theorem? I would suspect this question is asking you to find the continuous linear functional on R^4 (a Hilbert space) associated with u=(1,1,2,2) (u is used as in the notation on the planetmath website).
     
  4. Nov 21, 2007 #3
    So in the inner product what is x?

    What is definition of inner product.
     
  5. Nov 21, 2007 #4
    Wasn't clear in my last post:

    there were 2 questions

    1- So in the inner product (in planetmath.org) what is x?

    2- What is definition of inner product.
     
  6. Nov 21, 2007 #5

    morphism

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    The usual inner product on R^4 is the dot product.
     
  7. Nov 21, 2007 #6

    HallsofIvy

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    If I remember correctly, the "linear functional" associated with the given vector v in an inner product space is just f(x)= <v, x> where < , > is the inner product.
     
  8. Dec 1, 2007 #7
    Morphism - Do I just <v,v> for the linear functional. I don't clearly understand what the other term I need.

    Sharkie
     
  9. Dec 1, 2007 #8

    morphism

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    Is f(x) = <(1,2), x> a linear functional on R^2?
     
  10. Dec 2, 2007 #9
    Yes.. the way to prove is below

    Scalar addition

    f(u) = <(1,2), u> = 1*u + 2*u = u + 2u = 3u
    f(g) = <(1,2), g> = = 3g
    f(u+g) = <(1,2), (u,g)> = 1*u + 1*g + 2*u + 2*g = 3u + 3g

    f(u) + f(g) = f(u+g)

    Scalar multiplication

    f(kx) = <(1,2), kx> = kx*1 + 2*kx = 3kx
    kf(x) = k <(1,2), x> = k(1*x + 2*x) = 3kx

    Since it is closed under both scalar multiplication and additoin, it is a linear functional.

    But how does it help me in my actual problem ?
    Sorry I don't see the angle

    Sharkie
     
  11. Dec 2, 2007 #10

    HallsofIvy

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    Your question has been answered several times in these responses! The linear functional associated with vector v is f(x)= <v, x>.

    Oh, and since R4 is finite dimensional, talking about "Hilbert Spaces" and "Riesz representation" is overkill!
     
  12. Dec 6, 2007 #11
    OK - I think I was missing a key info and finally figured it out. On more reading, I realised that a linear functional maps into a scalar. Thats the key I was missing. And all the other exampls made sense then. However, this problem is not

    Consider the linear functional f:Rn --> R defined by f(x1,x2,. . .,xn)= x1 + x2 +. . . + xn. Find the vector u in Rn such that for all vectors v in Rn we have f(v)=(v,u), where ( , ) is the usual inner product.

    The reason it isn't make sense is where does vector x come into the picture.
     
  13. Dec 6, 2007 #12

    Office_Shredder

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    They're asking what vector you have to dot with x1,...,xn to get x1+...+xn
     
  14. Dec 6, 2007 #13
    Isn't that just the (1.......1) vector (size 1 x n)
     
  15. Dec 6, 2007 #14

    Office_Shredder

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