Inner product Pythagoras theorem

[motlking]

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

Thanks

 

[dextercioby]

The LHS is

$$\langle x_{1}+x_{2}+..., x_{1}+x_{2}+...\rangle$$

while the RHS is

$$\langle x_{1},x_{1}\rangle +\langle x_{2},x_{2}\rangle +...$$

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

$$\langle x_{i},x_{j} \rangle =0 \ \forall \ i\neq j$$

the equality follows easily.

Daniel.

 

[motlking]

Thanks a lot Daniel I feel a little silly, anyway wish me luck for my exam tomorrow!

 

[ObsessiveMathsFreak]

I think you meant, principal bundle.

A principle bundle is a bundle with a moral fibre.