Mathematical Physics Vol 1

1.5 Algebraic model of linear vector space

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1 ◦ addition is commutative

x + y = y + x ,

(1.48)

2 ◦ addition is associative

( x + y )+ z = x +( y + z ) ,

(1.49)

3 ◦ in X there exists a zero vector 0 , such that

x + 0 = x , for each x ∈ X , (1.50) 4 ◦ to each vector x ∈ X corresponds an opposite vector in X , denoted by − x , such that x +( − x )= 0 . (1.51) Multiplication is distributive 5 ◦ with respect to the addition of vectors α ( x + y )= α x + α y , (1.52) 6 ◦ and with respect to the addition of scalars ( α + β ) x = α x + β x , (1.53) 7 ◦ multiplication by a scalar is associative α ( β x )=( αβ ) x , (1.54) 8 ◦ multiplication of a vector by the scalar 1 leaves the vector unchanged i.e. 1 x = x , where 1 ∈ T . (1.55) Definition A vector space ( X , T ) is normed if there exists a nonnegative function ∥ x ∥ , defined for each x ∈ X , with the following properties ∥ 0 ∥ = 0 and ∥ x ∥ > 0 , for x̸ = 0 , (1.56) ∥ λ x ∥ = | λ |·∥ x ∥ , for each λ ∈ T , (1.57) ∥ x + y ∥≤∥ x ∥ + ∥ y ∥ (rule of the triangle) . (1.58) This function is called the norm of x . Definition Let X be a set whose elements are denoted by x , y ... . If to each ordered pair ( x , y ) from X a real number d ( x , y ) is assigned, with the following properties 0 ≤ d ( x , y ) < + ∞ , (1.59) d ( x , y )= 0 ⇔ x = y , (1.60) d ( x , y )= d ( y , x ) , (1.61) d ( x , y ) ≤ d ( x , z )+ d ( z , y ) , (1.62) then it is said that the set X is equipped with the metric d . Aset X equipped with the metric d is called a metric space . Its elements are called points, and d ( x , y ) is called the distance between points x and y . A metric space is, thus, a pair ( X , d ) .