Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.2 LINEAR OPERATORS

where the equality holds if the sum includes allNbasis vectors. If not

all the basis vectors are included in the sum then the inequality results

(though of course the equality remains if those basis vectors omitted all

haveai= 0). Bessel’s inequality can also be written

〈a|a〉≥

∑

i

|ai|^2 ,

where theaiare the components ofain the orthonormal basis. From (8.16)

these are given byai=〈eˆi|a〉. The above may be proved by considering

∣

∣

∣

∣

∣

∣

∣

∣a−

∑

i

〈ˆei|a〉ˆei

∣

∣

∣

∣

∣

∣

∣

∣

2

=

〈

a−

∑

i

〈ˆei|a〉eˆi

∣

∣

∣a−

∑

j

〈ˆej|a〉ˆej

〉

.

Expanding out the inner product and using〈eˆi|a〉∗=〈a|ˆei〉, we obtain

∣

∣

∣

∣

∣

∣

∣

∣a−

∑

i

〈eˆi|a〉eˆi

∣

∣

∣

∣

∣

∣

∣

∣

2

=〈a|a〉− 2

∑

i

〈a|ˆei〉〈eˆi|a〉+

∑

i

∑

j

〈a|eˆi〉〈ˆej|a〉〈ˆei|ˆej〉.

Now〈ˆei|ˆej〉=δij, since the basis is orthonormal, and so we find

0 ≤

∣

∣

∣

∣

∣

∣

∣

∣a−

∑

i

〈ˆei|a〉ˆei

∣

∣

∣

∣

∣

∣

∣

∣

2

=‖a‖^2 −

∑

i

|〈eˆi|a〉|^2 ,

which is Bessel’s inequality.

We take this opportunity to mention also

(iv) theparallelogram equality

‖a+b‖^2 +‖a−b‖^2 =2

(

‖a‖^2 +‖b‖^2

)

, (8.22)

which may be proved straightforwardly from the properties of the inner

product.

8.2 Linear operators

We now discuss the action oflinear operatorson vectors in a vector space. A

linear operatorAassociates with every vectorxanother vector

y=Ax,

in such a way that, for two vectorsaandb,

A(λa+μb)=λAa+μAb,

whereλ,μare scalars. We say thatA‘operates’ onxto give the vectory.We

note that the action ofAisindependentof any basis or coordinate system and