7.2.  Inner Product Spaces

We saw in the previous section that the notion of dot product on helps us to analyze many geometric concepts. As a vector space is the abstraction of the space with addition  and multiplication, the notion of dot product on also can be made abstract (not just for abstraction sake, it gives rich dividends, as we shall see later ). The aim of this section is to define the notion of dot product ( or inner product ) on any vector space. Given a vector space V  over , we can define the notion of inner-product as a map from V V such that it has the properties of the dot product on . And  using this, we define the useful notions of angle and distance in general inner-product spaces.

A map ( v, w)   from V V is called  an inner-product on V if it has the following
properties:

         (i) 

         (ii)   .

         (iii)   denotes the complex conjugate of  u .

         (iv)  

       A vector space V on which an inner-product is defined is called an inner-product space.

Note that (iii) implies that

In the property (iii) of the above definition, it is necessary to have complex conjugation in the right hand side.
If we just demand that , as for the standard dot product on , then we will have a
contradiction. For example, if that were the case, then using (iv) for every v V , v 0, we have

                                         ,

a contradiction. The property (iv) is equally important for it will define the 'magnitude' of a vector in an abstract inner-product space.

(i) Let be nonnegative real numbers. For and ,define

    It is easy to check that this is an inner-product on . When each , this is called the standard inner-product on .

(ii) On , as a vector space over , for and , define

                                           .

     Then this is an inner-product on and is called the standard inner-product on .

(iii) Let V = [a, b], the space of all real valued Riemann-integrable functions on [a, b]. For   f, g [a, b], u and x , define

                                    ( f + g ) (x) :=  f (x) + g (x),
                                             ( u f ) (x) := u f (x).

   With these operation of addition and scalar multiplication, [a, b]  becomes a vector space over .     This vector  space is not finite dimensional. For  f, g [a, b] define

                                                   .

One can visualize this as a generalization of the inner-product on as defined in example (i): imagine a function  f : [a, b] , to be a vector with infinite number of components, the xth component being  f(x), x [a, b]. Then, is the generalization of , with replaced by . From the standard theorems on Riemann integration, it follows that is a well-defined inner-product on           [a, b].

                  There are many such examples of inner-product space which arise in various problems in diverse fields. Notions of distance and angle in motivates the following :

(i)   The magnitude or the norm of a vector vV  is defined as

(ii)   For v, w V  we say v is perpendicular or orthogonal to w, if  . We also denote this as 
       by  ,

(iii)  A set of vectors S = V  is called an orthogonal set if

(iv)  A set of vector S = V  is called an orthonormal set if

(i)  In with the standard inner-product, as defined in example 7.2.4(i). For 1 i   n, let  , where 1 appears at the ith place. Let S = is an orthonormal set.

(ii)  In the inner-product space , as defined in example 7.2.4(iii), let   and . Then all the sets, both are orthonormal sets.

For u, v, w  V , the following holds :

(i)  Parallelogram identity :

(ii)  Pythagoras identity , then

(iii)  Cauchy-Schwarz's inequality and the equality holds iff { u, v } is a linearly
       dependent set.

                                                                                                                                                   Proof