Inner product Pythagoras theorem

  • Thread starter motlking
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  • #1
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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.

Theorem
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

Proof
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:

Thanks
 

Answers and Replies

  • #2
dextercioby
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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.

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

A principle bundle is a bundle with a moral fibre.
 

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