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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

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# Inner product Pythagoras theorem

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