1 Definition of an Inner Product Space

Section 4.1 Definition of an Inner Product Space

In all of our discussions in Chapters 1, 2 and 3, we have allowed the scalar field to be the set of complex numbers C . Most of our examples have utilized the special case of real numbers R . In this chapter we shall discuss a so called eigenvalue problem that arises in many areas of the applications. As we shall see, a discussion of this problems is best done in the context where the scalar field is allowed to be C . It is for this reason, we shall continue to allow complex numbers for the scalar field. Thus, we shall first define an inner product in this more general case. We will often illustrate the concepts with examples where the scalar field is the set of real numbers.

It is useful to briefly remind ourselves of certain arithmetic manipulations with complex numbers. You will recall that complex numbers always have the representation

Chap. 4 •

INNER PRODUCT SPACES   a ib (4.1.1)

where 2 a and b are real numbers and the symbol i obeys i   . The complex conjugate of a 1 complex number  is the complex number

   (4.1.2) a ib

It follows from (4.1.1) and (4.1.2) that

  a ib a ib 2 a  2 Re( )  (4.1.3)

where Re( )  means the real part of the complex number  . Also, if   a ib , then

2 2 2 2 2     ( a  ib a )(  ib )  a  iab  iab  ib  a  (4.1.4) b

is a real number. If one refers to the absolute value of a complex number it is a real number denoted by  and it is defined by

2 2     a  b (4.1.5)

When dealing with complex numbers, it is convenient to useful to have a formal condition which reflects the special case when a complex number is, in fact, real. This condition is the equation

  . In order to see that this condition does imply that  is real simply form the equality   . The result is

      (4.1.6) a ib  a ib

Equation (4.1.6) implies that b  0 , and thus the complex number  is real.

If you are given a vector space with its long list of properties summarized in Section 2.1, an inner product space is a vector space with additional properties that we shall now summarize.

Definition : An inner product on a complex vector space V is a function f : V  V C with the

following properties: (1)

f  uv ,  f vu ,

 f  uv ,  f   uv , 

f  u  wv ,   f  uv ,  f  wv , 

f  uu ,   0 and f  uu ,   0 if and only if u  0

Sec. 4.1 • Definition of an Inner Product Space

for all uvwV ,,  and  C . In Property 1 the bar denotes the complex conjugate. Properties 2 and 3 require that f be linear in its first argument; i.e.,

f   u   vw ,    f  uw ,    f  vw ,  (4.1.7)

for all uvwV ,,  and all ,   C . Property 1 and the linearity implied by Properties 2 and 3 insure that f is conjugate linear in its second argument; i.e.,

f  u ,  v   w    f  uv ,   f  uw ,  (4.1.8)

for all uvwV ,,  and all ,   C . Note that Property 1 shows that fuu  ,  is real. Property 4 requires that f be positive definite.

There are many notations for the inner product. In cases where the vector space is a real one, the notation of the “dot product” is used. In this case, one would write the function

f : V  V R as

f  uv ,  uv (4.1.9)

In cases where the vector space is a complex one, it is useful to adopt the notation

f  uv ,  uv , (4.1.10)

for the function f : V  V C . In this work, we shall adopt the notation (4.1.10).

An inner product space is simply a vector space with an inner product. To emphasize the importance of this idea and to focus simultaneously all its details, we restate the definition as follows.

Definition.

A complex inner product space, or simply an inner product space, is a set V and a field C such that:

(a) There exists a binary operation in V called addition and denoted by  such that: (1)

 u  v   w u  v  w  for all uvwV ,, 

There exists an element 0V  such that u  0 u for all uV 