- #1
motlking
- 2
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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
Thanks
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