Page 228 -

5.4. EUCLIDEAN SPACES 195

5.4.2. Complex Euclidean Space (Unitary Space)

5.4.2-1. Definition and properties of complex Euclidean space (unitary space).
A complex Euclidean space (or unitary space) is a complex linear space V endowed with
a scalar product (also called inner product), which is a complex-valued function of two
arguments x V and y V called their scalar product, denoted by x ⋅ y, satisfying the
following conditions (called axioms of the scalar product):
1. Commutativity: x ⋅ y = y ⋅ x.
2. Distributivity: (x 1 + x 2 ) ⋅ y = x 1 ⋅ y + x 2 ⋅ y.
3. Homogeneity: (λx) ⋅ y = λ(x ⋅ y) for any complex λ.
4. Positive definiteness: x ⋅ x ≥ 0;and x ⋅ x = 0 if and only if x = 0.
Here y ⋅ x is the complex conjugate of a number y ⋅ x.

n
Example 1. Consider the n-dimensional complex linear space R ∗ whose elements are ordered systems
of n complex numbers, x =(x 1, ... , x n). We obtain a unitary space if the scalar product of two elements
x =(x 1, ... , x n)and y =(y 1, .. . , y n) is introduced by
x ⋅ y = x 1 ¯y 1 + ·· · + x n ¯y n,
where ¯y j is the complex conjugate of y j.

THEOREM. For any two elements x and y of an arbitrary unitary space, the Cauchy–
Schwarz inequality holds:
2
|x ⋅ y| ≤ (x ⋅ x)(y ⋅ y).
THEOREM. Any unitary space becomes a normed space if the norm of its element x is
introduced by
√
x = x ⋅ x. (5.4.2.1)
COROLLARY. For any two elements x and y of a normed Euclidean space with the norm
(5.4.2.1), the triangle inequality (5.4.1.1) holds.
The distance between elements x and y of a unitary space is defined by

d(x, y)= x – y . (5.4.2.2)

Two elements x and y of a unitary space are said to be orthogonal if their scalar product
is equal to zero, x ⋅ y = 0.

5.4.2-2. Orthonormal basis in a finite-dimensional unitary space.

Elements x 1 , ... , x m of a unitary space V are linearly independent if and only if their Gram
determinant is different from zero, det[x i ⋅ x j ] ≠ 0.
One says that elements i 1 , ... , i n of an n-dimensional unitary space V form an or-
thonormal basis of that space if these elements are mutually orthogonal and have unit norm,
i.e.,
1 for i = j,
i i ⋅ i j =
0 for i ≠ j.
Given any n linearly independent elements of a unitary space, one can construct an
orthonormal basis of that space using the procedure described in Paragraph 5.4.1-2 (see
formulas (5.4.1.3)).