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Inner product Pythagoras theorem

  1. Nov 6, 2006 #1
    Hey guys,

    I am studying atm and looking at this book: "Introduction to Hilbert Space" by N.Young.

    For those who have the book, I am referring to pg 32, theorem 4.4.

    If x1,...,xn is an orthogonal system in an inner product space then,

    ||Sum(j=1 to n) xj ||^2 = Sum(j=1 to n) ||xj||^2

    Write the LHS as an inner product space and expand.

    Does anyone know what steps are needed to do this?

    This is what I have done:

    ||Sum(j=1 to n) xj ||^2 = ( Sum(j=1 to n) xj, Sum(j=1 to n) xj(conjugate))
    = Sum(j=1 to n) xjxj(conjugate)
    =Sum(j=1 to n) ||xj||^2 as requuired....

    Is this correct?

    Any help would be great for what should be an easy question :blushing:

  2. jcsd
  3. Nov 6, 2006 #2


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    The LHS is

    [tex] \langle x_{1}+x_{2}+..., x_{1}+x_{2}+...\rangle [/tex]

    while the RHS is

    [tex]\langle x_{1},x_{1}\rangle +\langle x_{2},x_{2}\rangle +... [/tex]

    and the 2 sums go up to "n". Since

    [tex] \langle x_{i},x_{j} \rangle =0 \ \forall \ i\neq j[/tex]

    the equality follows easily.

  4. Nov 6, 2006 #3
    Thanks alot Daniel :redface: I feel a little silly, anyway wish me luck for my exam tomorrow! :approve:
  5. Nov 6, 2006 #4
    I think you meant, principal bundle.

    A principle bundle is a bundle with a moral fibre.
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