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5.4. EUCLIDEAN SPACES 193

THEOREM. For any two elements x and y of a Euclidean space, the Cauchy–Schwarz
inequality holds:
2
(x ⋅ y) ≤ (x ⋅ x)(y ⋅ y).

A linear space V is called a normed space if it is endowed with a norm,which is a
real-valued function of x V, denoted by x and satisfying the following conditions:
1. Homogeneity: λx = |λ| x for any real λ.
2. Positive definiteness: x ≥ 0 and x = 0 if and only if x = 0.
3. The triangle inequality (also called the Minkowski inequality) holds for all elements
x and y:
x + y ≤ x + y . (5.4.1.1)
The value x is called the norm of an element x or its length.
THEOREM. Any Euclidean space becomes a normed space if the norm is introduced by
√
x = x ⋅ x. (5.4.1.2)
COROLLARY. In any Euclidean space with the norm (5.4.1.2), the triangle inequality
(5.4.1.1) holds for all its elements x and y.
The distance between elements x and y of a Euclidean space is defined by

d(x, y)= x – y .

One says that ϕ [0, 2π]is the angle between two elements x and y of a Euclidean
space if
x ⋅ y
cos ϕ = .
x y
Two elements x and y of a Euclidean space are said to be orthogonal if their scalar product
is equal to zero, x ⋅ y = 0.
PYTHAGOREAN THEOREM. Let x 1 , ... x m be mutually orthogonal elements of a Eu-
clidean space, i.e., x j ⋅ x j = 0 for i ≠ j.Then
2
2
2
x 1 + ··· + x m = x 1 + ··· + x m .
Example 3. In the Euclidean space B 3 of free vectors with the usual scalar product (see Example 1), the
following relations hold:
2 2 2
a = |a|, (a ⋅ b) ≤ |a| |b| , |a + b| ≤ |a| + |b|.
n
In the Euclidean space R of ordered systems of n numbers with the scalar product defined in Example 2,
the following relations hold:

2
2
x = x + ··· + x n ,
1
2 2 2 2 2
(x 1y 1 + ··· + x ny n) ≤ (x 1 + ··· + x n )(y 1 + ··· + y n ),

2
2
(x 1 + y 1) + ··· +(x n + y n) ≤ x 1 + ··· + x n y 1 + ··· + y n .
2 2 2 2
5.4.1-2. Orthonormal basis in a finite-dimensional Euclidean space.
For elements x 1 , ... , x m of a Euclidean space, the mth-order determinant det[x i ⋅ x j ]is
called their Gram determinant. These elements are linearly independent if and only if their
Gram determinant is different from zero.