Inner product Pythagoras theorem

  • Thread starter motlking
  • Start date
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
 

dextercioby

Science Advisor
Homework Helper
Insights Author
12,944
522
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.
 
Thanks alot Daniel :redface: I feel a little silly, anyway wish me luck for my exam tomorrow! :approve:
 
I think you meant, principal bundle.

A principle bundle is a bundle with a moral fibre.
 

Related Threads for: Inner product Pythagoras theorem

Replies
2
Views
782
  • Posted
Replies
10
Views
3K
  • Posted
Replies
1
Views
3K
Replies
2
Views
3K
Replies
15
Views
3K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top