Page 2 of 2 FirstFirst 12
Results 16 to 26 of 26

Math Help - how to find the inner product formulla

  1. #16
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374
    No, that is the inner product in Euclide space, not Cartesian product.
    the Norm is induced by the inner product as:
    |(r,f(x))|=\sqrt{|r|^2+\int_0^1|f(x)|^2 dx}
    Follow Math Help Forum on Facebook and Google+

  2. #17
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    do you know how to get to the formula?

    its <f,g>=

    what you written doesnt resemble this form
    and its not the correct answer
    Follow Math Help Forum on Facebook and Google+

  3. #18
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374
    The element of \mathbb{C}\times L^2 is not function, but a vector consisting of a complex number and a function. If you have problem about Cartesian Product, please turn to any function analysis book or set theory.
    It is impossible to define a norm for function which has the required form and property.
    I am sure I did not make any mistake, I've check it for several times.
    Do notice that: the given basis is not a orthogonal basis of the closed subspace.
    Follow Math Help Forum on Facebook and Google+

  4. #19
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    actually the answer is much simpler
    i was expexting something like this

    http://i45.tinypic.com/2jfhv9z.jpg

    out of one of your posts
    and as you see its nothing like your definition

    do you know how to get to this formula from the given norm?
    Last edited by transgalactic; November 21st 2009 at 07:50 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #20
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374
    well, "two Defferent paths, the same destination"
    Every route reaches Rome.
    You can give you comment on these two method after you get the final result, and please let me know if you get the same result ,OKA?
    I have been convinced for your solution ,I take back part of my words , finally thank you for your question and reply.
    Follow Math Help Forum on Facebook and Google+

  6. #21
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    there is no reply

    you didnt tell me how to get to this formula from this norm
    Follow Math Help Forum on Facebook and Google+

  7. #22
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374
    Think about it in a inverse direction, we can find some hint(or indication ) which lead to the given definition of the solution.
    further, My method can be generalized to a more general case, It extends the dimension of the Hilbert Space and the space itself.
    Follow Math Help Forum on Facebook and Google+

  8. #23
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    it seems you cant do it your self

    show equations
    from the given norm to inner product formula
    Follow Math Help Forum on Facebook and Google+

  9. #24
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374

    Two way leading to the same destination

    OK,I will do it in my way,then do it again in your way,and you will see that there is a clear corresponding between these two definition and method.
    My way to solve it:
    define the inner product as above mentioned In \mathbb{C}\times L^2, Then
    The problem is equivalent to "Find the optimal solution and the distance between point (1,3x^2) and the
    closed subspace spanned by three vectors (1,0),(1,1) and (1,2x) in \mathbb{C}\times L^{2}."
    By using Schimidt orthognalization, we get:  e_1=(1,0),e_2=(0,1) and
    e_3=\sqrt{3}(0,2x-1) are the orthogonal base in the colsed subspace spanned by those three vectors.
    let t=(1,3x^2), then the minimal point is
    <t,e_1>e_1+<t,e_2>e_2+<t,e_3>e_3 that is, (1,3x-\frac{1}{2}) Thus we get:
    \alpha+\beta+\gamma=1
    \beta+2\gamma x=3x-\frac{1}{2}
    which lead to \alpha=0,\beta=-\frac{1}{2},\gamma=\frac{3}{2} .
    Do it again in your way:
    define the inner product as your definition.
    The problem is equivalent to find the optimal solution and the distance between point x^3 and the closed subspace spanned by 1(constant function),x,x^2 in L^2.
    Since v_1=1,v_2=x-1,v_3=\sqrt{3}(x^2-x) are the orthogonal base in the closed subspace,(can you see the connection between these two basis,and the corresponding between these two methods),By using the Projection Theorem In Hilbert space, we get the same result: \alpha=0,\beta=-\frac{1}{2},\gamma=\frac{3}{2} .
    If you completely understand my way, you will see process from the norm to inner product, the f(1)\overline{g(1)} part in the definition just simplify the process and computaion.
    Follow Math Help Forum on Facebook and Google+

  10. #25
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    thanks for the alternative way

    how you came up with
    <br />
(1,3x^2)<br />

    i cant imagine this vector
    from where the 1 comes from?
    Follow Math Help Forum on Facebook and Google+

  11. #26
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374
    I thought it is impossible to define the inner product and norm in L^2 which has the required form and property, So I need to extend L^2 to a "bigger" space, and then define the inner product and norm. You can see your inner product is just a special case of my definition, Your Space is a closed subspace of the extended space.
    (1,3x^2) is the given point in the extended space, consider \alpha,\beta,\gamma as the coefficients of the linear combination of the three vectors, thus 1 comes from here.
    If you still have problems please review the Extended Space and its inner product definition(similar to R^n).
    Follow Math Help Forum on Facebook and Google+

Page 2 of 2 FirstFirst 12

Similar Math Help Forum Discussions

  1. Find the product
    Posted in the Algebra Forum
    Replies: 12
    Last Post: January 16th 2010, 04:36 PM
  2. Find the Product
    Posted in the Algebra Forum
    Replies: 6
    Last Post: April 21st 2009, 10:01 PM
  3. Find each product. please help :(
    Posted in the Algebra Forum
    Replies: 3
    Last Post: November 14th 2007, 04:59 PM
  4. please help :( Find the product.
    Posted in the Algebra Forum
    Replies: 1
    Last Post: November 14th 2007, 03:39 PM
  5. Find the product of...
    Posted in the Math Topics Forum
    Replies: 2
    Last Post: January 30th 2007, 07:42 PM

Search Tags


/mathhelpforum @mathhelpforum