Inner product Pythagoras theorem

In summary, the conversation discusses a theorem in the book "Introduction to Hilbert Space" by N.Young, specifically on page 32, theorem 4.4. The theorem states that if x1,...,xn is an orthogonal system in an inner product space, then the norm squared of the sum of the xi's is equal to the sum of the norms squared of each individual xi. The conversation also includes a proof for this theorem and clarifies the steps needed to expand the LHS as an inner product space. The conversation ends with a mention of a principle bundle, which is a bundle with a moral fibre.
  • #1
motlking
2
0
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
 
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  • #2
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
Thanks a lot 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.
 

Related to Inner product Pythagoras theorem

1. What is the "Inner product Pythagoras theorem"?

The "Inner product Pythagoras theorem" is a mathematical theorem that relates the inner product of two vectors to the length of those vectors. It states that the square of the length of the sum of two vectors is equal to the sum of the squares of the lengths of each individual vector, plus two times the inner product of the two vectors.

2. What is an inner product?

An inner product is a mathematical operation that takes two vectors as inputs and produces a scalar value as an output. It is often denoted by the symbol <x,y> and is used to measure the angle between two vectors and the length of a vector.

3. How is the inner product Pythagoras theorem used in real life?

The inner product Pythagoras theorem is used in various applications such as signal processing, image compression, and quantum mechanics. It is also used in physics and engineering to calculate the work done by a force and to determine the distance between two points in a multidimensional space.

4. Can the inner product Pythagoras theorem be extended to more than two vectors?

Yes, the inner product Pythagoras theorem can be extended to any number of vectors. It is known as the Law of Cosines and is used to find the length of a vector in a multidimensional space.

5. How is the inner product Pythagoras theorem related to the traditional Pythagorean theorem?

The traditional Pythagorean theorem relates the length of the sides of a right triangle, while the inner product Pythagoras theorem relates the lengths of vectors. The traditional Pythagorean theorem can be derived from the inner product Pythagoras theorem by setting one of the vectors to be perpendicular to another.

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