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         INNER PRODUCT AND INNER PRODUCT SPACE

The inner product as defined for abstract vector spaces is an abstracted form of the dot (or scalar) product of complex n-dimensional Euclidean space. It is the abstract vector space equivalent of the dot product of complex n-space V_n(C). It is an opaque, axiomatic one. As in the definitions of many other concepts of abstract mathematics a set of axiomatic requirements are stated that must be met. These requirements or postulates derive from a list of properties of the inner product of complex n-dimensional Euclidean space. Before giving the definition let us review the properties of the dot product of complex n-space V_n(C).

Conjugate of a vector. If X is a vector having complex numbers as elements, the vector obtained from X by replacing each element by its conjugate is called the conjugate of X and is denoted by i.e. the conjugate of the vector

is

            .

Inner (or dot or scalar) product of two complex n-vectors. Let

and

be two vectors whose elements are complex numbers. Then their inner product is given by

Laws governing inner products of complex n-vectors. Let X, Y and Z be complex n-vectors and c be a complex number. Then the following laws hold:

1.

2.

3.

4.

5.

6. where is the real part of

7. where is the complex part of

Def. Inner Product. Let V be an abstract vector space over the field F where F is either the field of real or complex numbers. An inner product on V is a function which assigns to each ordered pair of vectors x, y in V a scalar in F, called the inner product x•y, such that the following axioms hold  

The inner product of two vectors is usually denoted by the notation “(x, y)” instead of the dot notation x•y. In this notation the above four axioms are

     where c is a scalar and the over-bar denotes complex conjugation

Def. Inner product space. A vector space on which an inner product is defined.