7.2 Sets and Spaces
393
Since
|x n + y n |
p
(|x n | + |y n |)
p
2
p
[max(|x n |, |y n |)]
p
2
p
(|x n |
p
+ |y n |
p
)
there is
∞
n=1
|x n + y n |
p
2
p
∞
n=1
|x n |
p
+
∞
n=1
|y n |
p
< ∞
hence x + y ∈ l
p , l
p is a linear space.
In pth power summable sequence space, when p is equal to 2, it is the Hilbert
sequential space, which is called the square-summable sequence space, it is denoted
by l
2 , of course it is also a linear space. This space was introduced and researched
by Hilbert in 1912.
Let X be a linear space in the number field P, E ⊂ X , if there exists arbitrary x,
y ∈ E, λ ∈ [0, 1], there is λx + (1 − λ)y ∈ E, then E is called the convex set. The
Geometric meaning of convex set is that the connecting line between arbitrary two
points in convex set is still in the set.
Let X be a linear space in the number field P, if for every element x in X , according
to certain rules, there corresponds the determined nonnegative real number it is
denoted by x or •, and the following three conditions (axioms of the norm) are
simultaneously satisfied, namely for any x, y ∈ X , α ∈ P, there are (1) the positive
definiteness: x 0, and the necessary and sufficient conditions of x = 0 is
x = 0; (2) the homogeneity: αx = |α||x; (3) the triangle inequality: x + y
x + y, the x denotes the length of the element x, it is called the norm of x
on X . X and • all together are called the normed linear space, normed vector
space or linear normed space, it is denoted by (X, •). Sometimes X is also
called the normed linear space, normed vector space or linear normed space. In
normed linear space, the distance between the elements x and y can be defined by
ρ(x, y) = x − y. The norm is a generalization of the concept of vector length in
the Euclidean space. Obviously, any normed linear space shall be a metric space, it
is that the above defined distance satisfies distance axioms, that is
(1) ρ(x, y) = x − y 0, and the necessary and sufficient conditions of
ρ(x, y) = 0 is x = y;
(2) ρ(x, y) = x − y = −1(y − x) = y − x = ρ(y, x);
(3) ρ(x, z) = x − z + y − y x − y + y − z = ρ(x, y) + ρ(y, z).
Let R
n be n-dimensional linear space, for an arbitrary x ∈ R
n , • E can be
expressed as
• E = (x
2
1 + x
2
2 + · · · + x
2
n )
1
2
then • E is the norm on R
n , the norm is called the Euclidean norm on R
n .
393
Since
|x n + y n |
p
(|x n | + |y n |)
p
2
p
[max(|x n |, |y n |)]
p
2
p
(|x n |
p
+ |y n |
p
)
there is
∞
n=1
|x n + y n |
p
2
p
∞
n=1
|x n |
p
+
∞
n=1
|y n |
p
< ∞
hence x + y ∈ l
p , l
p is a linear space.
In pth power summable sequence space, when p is equal to 2, it is the Hilbert
sequential space, which is called the square-summable sequence space, it is denoted
by l
2 , of course it is also a linear space. This space was introduced and researched
by Hilbert in 1912.
Let X be a linear space in the number field P, E ⊂ X , if there exists arbitrary x,
y ∈ E, λ ∈ [0, 1], there is λx + (1 − λ)y ∈ E, then E is called the convex set. The
Geometric meaning of convex set is that the connecting line between arbitrary two
points in convex set is still in the set.
Let X be a linear space in the number field P, if for every element x in X , according
to certain rules, there corresponds the determined nonnegative real number it is
denoted by x or •, and the following three conditions (axioms of the norm) are
simultaneously satisfied, namely for any x, y ∈ X , α ∈ P, there are (1) the positive
definiteness: x 0, and the necessary and sufficient conditions of x = 0 is
x = 0; (2) the homogeneity: αx = |α||x; (3) the triangle inequality: x + y
x + y, the x denotes the length of the element x, it is called the norm of x
on X . X and • all together are called the normed linear space, normed vector
space or linear normed space, it is denoted by (X, •). Sometimes X is also
called the normed linear space, normed vector space or linear normed space. In
normed linear space, the distance between the elements x and y can be defined by
ρ(x, y) = x − y. The norm is a generalization of the concept of vector length in
the Euclidean space. Obviously, any normed linear space shall be a metric space, it
is that the above defined distance satisfies distance axioms, that is
(1) ρ(x, y) = x − y 0, and the necessary and sufficient conditions of
ρ(x, y) = 0 is x = y;
(2) ρ(x, y) = x − y = −1(y − x) = y − x = ρ(y, x);
(3) ρ(x, z) = x − z + y − y x − y + y − z = ρ(x, y) + ρ(y, z).
Let R
n be n-dimensional linear space, for an arbitrary x ∈ R
n , • E can be
expressed as
• E = (x
2
1 + x
2
2 + · · · + x
2
n )
1
2
then • E is the norm on R
n , the norm is called the Euclidean norm on R
n .