388
7 Variational Principles
Let f : X → Y be an injection, x ∈ X , if for an arbitrary y ∈ f (X ), there
corresponds unique x, the the mapping from f (X ) to X is called the inverse mapping
of f , it is written as f
−1 . At the moment, f
−1
[ f (x)] = f
−1
(y) = x, f [ f
−1
(y)] =
f (x) = y. f
−1
(y) namely x is called the inverse image of y under the mapping f .
The inverse mapping is the extension of the concept of inverse function.
Let there be two mappings f : X → Y 1 , g : Y 2 → Z , where Y 1 ⊂ Y 2 , if for
each x ∈ X , there exists h(x) = g[ f (x)], which determines a mapping from X to
Z h : X → Z , then h is called the composite mapping of f and g, it is denoted
by g ◦ f , that is h = g ◦ f , g ◦ f : X → Z or g ◦ f (x) = g[ f (x)], x ∈ X . The
composite mapping is a generalization of the concept of composite function. The
condition that the mappings f and g constitute composite mapping is that the range
of f must be included in the domain of g.
Let X be a nonempty set, if for each x ∈ X , there is the mapping I (x) = x, then
I : X → X is called the identity mapping on X . Obviously, the identity mapping
is a bijection.
Let there be two sets X and Y , if for each in advance given arbitrarily small
positive number ε, there exists a positive number δ, such that for all x suitable
for inequality 0 < |x − x 0 | < δ in X , the corresponding mapping f satisfies the
inequality |y − y 0 | = | f (x) − f (x 0 )| < ε, then f is called the continuous mapping
on X .
If there is a bijection f between two sets X and Y , then the two sets are called
the equipollence, equivalence or equivalent, they are denoted by X ~ Y or X
f
∼ Y .
Equipollence is that the numbers of the elements in two sets is equal. If a set and
the set of natural numbers are equipollent, then the set is called the countable
set, (de)numerable set or enumerable set. The set of natural numbers itself is
a countable set. The equivalence relation has the following three properties.
(1) Reflexive: X ~X .
(2) Symmetry: If X ~Y , then Y ~X .
(3) Transitivity: If X ~Y , Y ~Z , then X ~Z .
Let P be the set composed of some complex, there including 0 and 1, If performing
addition, subtraction, multiplication and division (divisor is not zero) operation to
arbitrary two numbers in P, the result is still the numbers in P, the P is called an
algebraic number field, number domain or number field.
7.2 Sets and Spaces
In elementary mathematics, the distance originally refers to the length of the line
between two points, the space originally denotes solid extent, they are both geometric
concepts, these two concepts in modern mathematics are very useful too, they have
a wider significance.
For a set, the elements in the set are determined, this set is completely determined,
but any relationship between the elements does not determined. In order to the practical need of studying a problem, a set can be introduced some different determined
7 Variational Principles
Let f : X → Y be an injection, x ∈ X , if for an arbitrary y ∈ f (X ), there
corresponds unique x, the the mapping from f (X ) to X is called the inverse mapping
of f , it is written as f
−1 . At the moment, f
−1
[ f (x)] = f
−1
(y) = x, f [ f
−1
(y)] =
f (x) = y. f
−1
(y) namely x is called the inverse image of y under the mapping f .
The inverse mapping is the extension of the concept of inverse function.
Let there be two mappings f : X → Y 1 , g : Y 2 → Z , where Y 1 ⊂ Y 2 , if for
each x ∈ X , there exists h(x) = g[ f (x)], which determines a mapping from X to
Z h : X → Z , then h is called the composite mapping of f and g, it is denoted
by g ◦ f , that is h = g ◦ f , g ◦ f : X → Z or g ◦ f (x) = g[ f (x)], x ∈ X . The
composite mapping is a generalization of the concept of composite function. The
condition that the mappings f and g constitute composite mapping is that the range
of f must be included in the domain of g.
Let X be a nonempty set, if for each x ∈ X , there is the mapping I (x) = x, then
I : X → X is called the identity mapping on X . Obviously, the identity mapping
is a bijection.
Let there be two sets X and Y , if for each in advance given arbitrarily small
positive number ε, there exists a positive number δ, such that for all x suitable
for inequality 0 < |x − x 0 | < δ in X , the corresponding mapping f satisfies the
inequality |y − y 0 | = | f (x) − f (x 0 )| < ε, then f is called the continuous mapping
on X .
If there is a bijection f between two sets X and Y , then the two sets are called
the equipollence, equivalence or equivalent, they are denoted by X ~ Y or X
f
∼ Y .
Equipollence is that the numbers of the elements in two sets is equal. If a set and
the set of natural numbers are equipollent, then the set is called the countable
set, (de)numerable set or enumerable set. The set of natural numbers itself is
a countable set. The equivalence relation has the following three properties.
(1) Reflexive: X ~X .
(2) Symmetry: If X ~Y , then Y ~X .
(3) Transitivity: If X ~Y , Y ~Z , then X ~Z .
Let P be the set composed of some complex, there including 0 and 1, If performing
addition, subtraction, multiplication and division (divisor is not zero) operation to
arbitrary two numbers in P, the result is still the numbers in P, the P is called an
algebraic number field, number domain or number field.
7.2 Sets and Spaces
In elementary mathematics, the distance originally refers to the length of the line
between two points, the space originally denotes solid extent, they are both geometric
concepts, these two concepts in modern mathematics are very useful too, they have
a wider significance.
For a set, the elements in the set are determined, this set is completely determined,
but any relationship between the elements does not determined. In order to the practical need of studying a problem, a set can be introduced some different determined