392
7 Variational Principles
belong to E. If x ∈ E, but x is not an accumulation point of E, then x is called the
isolated point of E. A set composed of all the accumulation points of E is called
the derived set of E, it is denoted by E
or E
d . A set composed of all the interior
points of E is called the interior of E, it is denoted by I nt E or E
0 . A set composed
of all the exterior points of E is called the exterior of E, it is denoted by E
e . The
union set of E and its interior is called the closure of E, it is denoted by E, namely
E = E ∪ E. If each point in E is its interior point, then E is called the open set.
If E contains all of its accumulation points, then E is called the closed set. If E is a
closed set without isolated point, then E is called the complete set or perfect set of
(X, ρ).
Let X be a linear set over the real number field, if there exists an element θ ∈ X
such that u + θ = u for every u ∈ X , then the element θ is called the zero element
or null element of the linear set X .
Let X be a nonempty set, arbitrary elements x, y, z ∈ X , θ is a zero element in X ,
it is called the identity element or unit element, P is a number field, α and β are
arbitrary numbers in P, if the additive operation is introduced in X , the multiplicative
operation is introduced in X and P, and the additive operation and multiplicative
operation satisfy the following rules: (1) x + y = y + x; (2) (x + y)+ z = x +(y + z);
(3) x + θ = x; (4) x + (−x) = θ , there −x is called the negative element or inverse
element of x; (5) 1α = α; (6) (αβ)x = α(βx); (7) (α + β)x = αx + βx; (8)
α(x + y) = αx + αy, then X is called the linear space or vector space on P. The
elements in X is called the vector. When P is a real number field or complex number
field, X is called the real linear space or complex linear space respectively. The
addition operation and the multiplication operation are generally called the linear
operation or linear space structure. Thus, the so-called linear space is the space
in which addition and multiplication operations can be performed to the elements,
Cartesian space for example is a linear space.
The set composed of all of the number sequence {x n }
∞
n=1 = {x 1 , x 2 , · · · , x n , · · · }
satisfying the condition
∞
n=1 |x n |
p
< ∞ in the number field M is called the pth
power summable sequence space, it is denoted by l
p , there 1 p < ∞. l
p can be
expressed as
l
p
= {x = { x n }
∞
n=1
x n ∈ M, n ∈ N with
∞
n=1
|x n |
p
< ∞}
Let an arbitrary number α belong to the number field M, for arbitrary x = {x n }
∞
n=1 ,
y = {y n }
∞
n=1 ∈ l
p , then there are
x + y = {x n + y n }
∞
n=1 , αx = {αx n }
∞
n=1 , αx ∈ l
p
7 Variational Principles
belong to E. If x ∈ E, but x is not an accumulation point of E, then x is called the
isolated point of E. A set composed of all the accumulation points of E is called
the derived set of E, it is denoted by E
or E
d . A set composed of all the interior
points of E is called the interior of E, it is denoted by I nt E or E
0 . A set composed
of all the exterior points of E is called the exterior of E, it is denoted by E
e . The
union set of E and its interior is called the closure of E, it is denoted by E, namely
E = E ∪ E. If each point in E is its interior point, then E is called the open set.
If E contains all of its accumulation points, then E is called the closed set. If E is a
closed set without isolated point, then E is called the complete set or perfect set of
(X, ρ).
Let X be a linear set over the real number field, if there exists an element θ ∈ X
such that u + θ = u for every u ∈ X , then the element θ is called the zero element
or null element of the linear set X .
Let X be a nonempty set, arbitrary elements x, y, z ∈ X , θ is a zero element in X ,
it is called the identity element or unit element, P is a number field, α and β are
arbitrary numbers in P, if the additive operation is introduced in X , the multiplicative
operation is introduced in X and P, and the additive operation and multiplicative
operation satisfy the following rules: (1) x + y = y + x; (2) (x + y)+ z = x +(y + z);
(3) x + θ = x; (4) x + (−x) = θ , there −x is called the negative element or inverse
element of x; (5) 1α = α; (6) (αβ)x = α(βx); (7) (α + β)x = αx + βx; (8)
α(x + y) = αx + αy, then X is called the linear space or vector space on P. The
elements in X is called the vector. When P is a real number field or complex number
field, X is called the real linear space or complex linear space respectively. The
addition operation and the multiplication operation are generally called the linear
operation or linear space structure. Thus, the so-called linear space is the space
in which addition and multiplication operations can be performed to the elements,
Cartesian space for example is a linear space.
The set composed of all of the number sequence {x n }
∞
n=1 = {x 1 , x 2 , · · · , x n , · · · }
satisfying the condition
∞
n=1 |x n |
p
< ∞ in the number field M is called the pth
power summable sequence space, it is denoted by l
p , there 1 p < ∞. l
p can be
expressed as
l
p
= {x = { x n }
∞
n=1
x n ∈ M, n ∈ N with
∞
n=1
|x n |
p
< ∞}
Let an arbitrary number α belong to the number field M, for arbitrary x = {x n }
∞
n=1 ,
y = {y n }
∞
n=1 ∈ l
p , then there are
x + y = {x n + y n }
∞
n=1 , αx = {αx n }
∞
n=1 , αx ∈ l
p