Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MATRICES AND VECTOR SPACES

8.1.3 Some useful inequalities

For a set of objects (vectors) forming a linear vector space in which〈a|a〉≥0for

alla, the following inequalities are often useful.

(i)Schwarz’s inequalityis the most basic result and states that

|〈a|b〉| ≤ ‖a‖‖b‖, (8.19)

where the equality holds whenais a scalar multiple ofb,i.e.whena=λb.

It is important here to distinguish between theabsolute valueof a scalar,

|λ|,andthenormof a vector,‖a‖. Schwarz’s inequality may be proved by

considering

‖a+λb‖^2 =〈a+λb|a+λb〉

=〈a|a〉+λ〈a|b〉+λ∗〈b|a〉+λλ∗〈b|b〉.

If we write〈a|b〉as|〈a|b〉|eiαthen

‖a+λb‖^2 =‖a‖^2 +|λ|^2 ‖b‖^2 +λ|〈a|b〉|eiα+λ∗|〈a|b〉|e−iα.

However,‖a+λb‖^2 ≥0 for allλ,sowemaychooseλ=re−iαand require

that, for allr,

0 ≤‖a+λb‖^2 =‖a‖^2 +r^2 ‖b‖^2 +2r|〈a|b〉|.

This means that the quadratic equation inrformed by setting the RHS

equal to zero must have no real roots. This, in turn, implies that

4 |〈a|b〉|^2 ≤ 4 ‖a‖^2 ‖b‖^2 ,

which, on taking the square root (all factors are necessarily positive) of

both sides, gives Schwarz’s inequality.

(ii) Thetriangle inequalitystates that

‖a+b‖≤‖a‖+‖b‖ (8.20)

and may be derived from the properties of the inner product and Schwarz’s

inequality as follows. Let us first consider

‖a+b‖^2 =‖a‖^2 +‖b‖^2 +2 Re〈a|b〉≤‖a‖^2 +‖b‖^2 +2|〈a|b〉|.

Using Schwarz’s inequality we then have

‖a+b‖^2 ≤‖a‖^2 +‖b‖^2 +2‖a‖‖b‖=(‖a‖+‖b‖)^2 ,

which, on taking the square root, gives the triangle inequality (8.20).

(iii)Bessel’s inequalityrequires the introduction of an orthonormal basiseˆi,

i=1, 2 ,...,Ninto theN-dimensional vector space; it states that

‖a‖^2 ≥

∑

i

|〈eˆi|a〉|^2 , (8.21)