404
7 Variational Principles
Proof (1) Let S n =
n
i=1 α i e i , σ n =
n
i=1 |α i |
2 , since A is the orthogonal system,
for any positive integers m and n, when n > m, from Eq. (7.3.1), there is
S n − S m
2
= α m+1 e m+1 + α m+2 e m+2 + · · · + α n e n
2
=
n
i=m+1
|α i |
2
= σ n − σ m
The necessary and sufficient conditions that the series {S n } is the Cauchy sequence
in H is that {σ n } is Cauchy sequence, according to the completeness of H and number
field, the inference (1) holds;
(2) Making the inner product of x =
∞
i=1 α i e i and e i , there is (x, e i ) =
∞
i=1 α i e i , e i
= α i (e i , e i ) = α i , namely α i = (x, e i ), then substituting it into
x =
∞
i=1 α i e i , we get x =
∞
i=1 (x, e i )e i ;
(3) According to Bessel inequality, the series
∞
i=1 |(x, e i )|
2
x
2 converges,
then from the inference (1) and inference (2), the series
∞
i=1 (x, e i )e i converges.
Quod erat demonstrandum.
7.4 Operators and Functionals
Let X and Y be two normed linear space in the same number field P, D is a subset of
X , if there exists is a corresponding rule T , such that for any x ∈ D, there corresponds
unique determined y = T (x) = T x ∈ Y , then T is called the operator or mapping
of D to Y in X , D is called the domain of definition or domain of T , it is denoted
by D(T ). y or T (x) is called the image of x, the set of an image { y|y = T x, x ∈ D}
is called the range or value field of T , it is denoted by T (D) or T D.
According to the different cases of the sets X , Y , in different branches of mathematics, the operator T has different conventional names. Conventionally, when X
and Y are both the number space, T is called the function; When X and Y are both
the normed linear spaces, as defined above, T is called the operator; When X is a
normed linear space, Y is a number space, T is called the functional, or the operator
that the range is a set of numbers is called the functional. When X is number space,
Y is a formed linear space, T is called the abstract function.
Let the domain of definition of the operator T be D, the range is T (D), u ∈ D,
f ∈ T (D), the equality T u = f is called the operator equation. Where, u is the
unknown function required. f is called the free term, it represents a source or sink.
When the free term f is zero, the operator is called the homogeneous equation.
Let u = φ be the boundary condition of an operator equation, where φ is a given
function on the boundary of the domain of definition D, if φ = 0, then the boundary
condition is called the homogeneous boundary condition.
Let X and Y be both the normed linear space, D is a subset of X , T is an operator
of D to Y , α is number in the number field P, then the following definitions can be
elicited by:
7 Variational Principles
Proof (1) Let S n =
n
i=1 α i e i , σ n =
n
i=1 |α i |
2 , since A is the orthogonal system,
for any positive integers m and n, when n > m, from Eq. (7.3.1), there is
S n − S m
2
= α m+1 e m+1 + α m+2 e m+2 + · · · + α n e n
2
=
n
i=m+1
|α i |
2
= σ n − σ m
The necessary and sufficient conditions that the series {S n } is the Cauchy sequence
in H is that {σ n } is Cauchy sequence, according to the completeness of H and number
field, the inference (1) holds;
(2) Making the inner product of x =
∞
i=1 α i e i and e i , there is (x, e i ) =
∞
i=1 α i e i , e i
= α i (e i , e i ) = α i , namely α i = (x, e i ), then substituting it into
x =
∞
i=1 α i e i , we get x =
∞
i=1 (x, e i )e i ;
(3) According to Bessel inequality, the series
∞
i=1 |(x, e i )|
2
x
2 converges,
then from the inference (1) and inference (2), the series
∞
i=1 (x, e i )e i converges.
Quod erat demonstrandum.
7.4 Operators and Functionals
Let X and Y be two normed linear space in the same number field P, D is a subset of
X , if there exists is a corresponding rule T , such that for any x ∈ D, there corresponds
unique determined y = T (x) = T x ∈ Y , then T is called the operator or mapping
of D to Y in X , D is called the domain of definition or domain of T , it is denoted
by D(T ). y or T (x) is called the image of x, the set of an image { y|y = T x, x ∈ D}
is called the range or value field of T , it is denoted by T (D) or T D.
According to the different cases of the sets X , Y , in different branches of mathematics, the operator T has different conventional names. Conventionally, when X
and Y are both the number space, T is called the function; When X and Y are both
the normed linear spaces, as defined above, T is called the operator; When X is a
normed linear space, Y is a number space, T is called the functional, or the operator
that the range is a set of numbers is called the functional. When X is number space,
Y is a formed linear space, T is called the abstract function.
Let the domain of definition of the operator T be D, the range is T (D), u ∈ D,
f ∈ T (D), the equality T u = f is called the operator equation. Where, u is the
unknown function required. f is called the free term, it represents a source or sink.
When the free term f is zero, the operator is called the homogeneous equation.
Let u = φ be the boundary condition of an operator equation, where φ is a given
function on the boundary of the domain of definition D, if φ = 0, then the boundary
condition is called the homogeneous boundary condition.
Let X and Y be both the normed linear space, D is a subset of X , T is an operator
of D to Y , α is number in the number field P, then the following definitions can be
elicited by: