Definition 1.4.1.
A vector space X over the field K is called a space with
scalar product if to each pair of elements 
there corresponds a certain number 
called the scalar product of the vectors 
and 
such that following condition (the axioms of scalar product) are satisfied:
2. 
, when 
is the conjugate complex number of 
;
3. 
(additivity with respect to the first factor);
4. 
(homogencity with respect to the first factor).
If 
is a vector space over 
,
then, by the definition, 
and condition 1 acquires the form 
,
i.e., in this case scalar product is commutative.
Example 1.4.1. Let us define in 
the scalar product of vectors
by the formula
Let us check the validity of conditions 1-4:
Example 1.4.2.Let us consider the vector
space 
of all functions integrable (in Lebesque's sense) on the interval 
We define the scalar product for
such functions by the formula
Verify that all the axioms 1-4 of scalar
product are satisfied.
Proposition 1.4.1. Scalar
product 
has the following properties:
1. 
(additivity with respect to the second factor);
2. 
(conjugate
homogeneity with respect to the second factor);
3. 
4. 
Let us prove these assertions:
Proposition
1.4.2 (Cauchy-Schwartz inequality). For arbitrary vectors
and 
of the vector space with scalar
product
it holds the inequality
Proof. If 
,
then, by the definition of the scalar
product (condition 1) the inequality holds. Now let us consider the
case 
We define an auxiliary function
As for 
The last inequality is equivalent to the inequality 
and this -- to the Cauchy-Schwartz
inequality. The Cauchy-Schwartz inequality makes it possible to define
the angle between two vectors by the scalar
product.
Definition
1.4.2. The angle between arbitrary vectors
and
of the vector space with scalar product
is defined by the formula
Problem 1.4.1.* Show that for each two
complex vectors
and
it holds the equality
Problem 1.4.2.* The scalar product in
the vector space 
of polynomials of at most degree n with real coefficients on 
is defined by the formula
Find the angle between the polynomials 
and 
