Mathematical Tools for Physics

6—Vector Spaces 156

on the domain−L < x <+Lis just an example of Fourier series, and the components offin this basis are
Fourier coefficientsa 1 ,...,b 0 ,.... An equally valid and more succinctly stated basis is

enπix/L, n= 0, ± 1 , ± 2 , ...

Chapter 5 on Fourier series shows many other choices of bases, all orthogonal, but not usually orthonormal.

6.8 Gram-Schmidt Orthogonalization
From a basis that is not orthonormal, it is possible to construct one that is. This device is called the Gram-Schmidt
procedure. Suppose that a basis is known (finite or infinite),~v 1 , ~v 2 ,...

Step 1: Normalize~v 1 : ~e 1 =~v 1

/√〈

~v 1 ,~v 1

〉

.

Step 2: Construct a linear combination of~v 1 and~v 2 that is orthogonal to~v 1 :
Let~e 20 =~v 2 −~e 1

〈

~e 1 ,~v 2

〉

and then normalize it.

~e 2 =~e 20

/〈

~e 20 ,~e 20

〉 1 / 2

.

Step 3: Let~e 30 =~v 3 −~e 1

〈

~e 1 ,~v 3

〉

−~e 2

〈

~e 2 ,~v 3

〉

etc.repeating step 2.
What does this look like? See problem 3.

6.9 Cauchy-Schwartz inequality
For common three-dimensional vector geometry, it is obvious that for any real angle,cos^2 θ≤ 1. In terms of a
dot product, this is|A~.B~|≤AB. This can be generalized to any scalar product on any vector space:
∣
∣

〈

~u,~v

〉∣

∣≤‖~u‖‖~v‖. (14)

The proof starts from a simple but not-so-obvious point. The scalar product of a vector with itself is by definition
positive, so for any two vectors~uand~vyou have the inequality

〈

~u−λ~v,~u−λ~v

〉

≥ 0. (15)

whereλis any complex number. This expands to

〈

~u,~u

〉

+|λ|^2

〈

~v,~v

〉

−λ

〈

~u,~v

〉

−λ*

〈

~v,~u

〉

≥ 0. (16)