394
7 Variational Principles
In the n-dimensional linear space the following important norms are often quoted
x ∞ = max
1in
|x i | [x = (x 1 , x 2 , · · · , x n ) ∈ R]
(7.2.5)
x 1 =
n
i=1
|x i |
(7.2.6)
x 2 =
n
i=1
|x i |
2
1
2
(7.2.7)
x p =
n
i=1
|x i |
p
1
p
(1 p < ∞)
(7.2.8)
Equation (7.2.5) is called the maximum norm or l ∞ norm, Eq. (7.2.6) is called
the sum norm or l 1 norm, Eq. (7.2.7) is called the Euclidean norm or l 2 norm,
Eq. (7.2.8) is called the Hölder norm or l p norm. Which can be verified that they
all satisfy the three axioms of norm, Eqs. (7.2.6)–(7.2.7) are the cases in which p in
Eq. (7.2.8) is equal to 1 and 2 respectively. Using the Cauchy inequality, it can be
proved that they satisfies the triangle inequality. Using the Minkowski inequality, it
can also be proved that Eq. (7.2.8) satisfies the triangle inequality.
Let X be a formed linear space, if any fundamental sequence in X is a convergent
sequence, then X is called the complete normed linear space, it is also called the
Banach space. The Banach space is one of the most important spaces in functional
analysis.
Let X be n-dimensional linear space in the number field P, x, y, z are arbitrary
vectors in X , α ∈ P, if the binary function (x, y) in X is introduced, and it has the
following properties: (1) The conjugate symmetry: When P is a real number field,
(x, y) = (y, x), or when P is complex number field, (x, y) = (y, x), there (y, x) the
conjugate complex number of (x, y); (2) The homogeneity: (αx, y) = α(x, y); (3)
The linearity or additivity: (x+y, z) = (x, z)+(y, z); (4) The positive definiteness:
(x, x) 0, and the necessary and sufficient conditions of (x, x) = 0 are x = 0,
then X is called the inner product space, (x, y) is called the inner product of the
vectors x and y. An inner product is also commonly denoted by the notation (·, ·). In
some books, an inner product can also be denoted by < x, y >, x · y and ( x|y) etc.
The inner product space is a special normed linear space. When the number field P
is a complex number field C, the inner product space X is called the complex inner
product space with the inner product (·, ·). When the number field P is a real number
field R, the inner product space X is called the real inner product space with the
inner product (·, ·). At this time, X is called the n-dimensional Euclidean space
about the real inner product, it is called the Euclidean space for short, it is denoted
by E
n . For example, the real space is three-dimensional Euclidean space, a plane
is two-dimensional Euclidean space, a straight line is one-dimensional Euclidean
7 Variational Principles
In the n-dimensional linear space the following important norms are often quoted
x ∞ = max
1in
|x i | [x = (x 1 , x 2 , · · · , x n ) ∈ R]
(7.2.5)
x 1 =
n
i=1
|x i |
(7.2.6)
x 2 =
n
i=1
|x i |
2
1
2
(7.2.7)
x p =
n
i=1
|x i |
p
1
p
(1 p < ∞)
(7.2.8)
Equation (7.2.5) is called the maximum norm or l ∞ norm, Eq. (7.2.6) is called
the sum norm or l 1 norm, Eq. (7.2.7) is called the Euclidean norm or l 2 norm,
Eq. (7.2.8) is called the Hölder norm or l p norm. Which can be verified that they
all satisfy the three axioms of norm, Eqs. (7.2.6)–(7.2.7) are the cases in which p in
Eq. (7.2.8) is equal to 1 and 2 respectively. Using the Cauchy inequality, it can be
proved that they satisfies the triangle inequality. Using the Minkowski inequality, it
can also be proved that Eq. (7.2.8) satisfies the triangle inequality.
Let X be a formed linear space, if any fundamental sequence in X is a convergent
sequence, then X is called the complete normed linear space, it is also called the
Banach space. The Banach space is one of the most important spaces in functional
analysis.
Let X be n-dimensional linear space in the number field P, x, y, z are arbitrary
vectors in X , α ∈ P, if the binary function (x, y) in X is introduced, and it has the
following properties: (1) The conjugate symmetry: When P is a real number field,
(x, y) = (y, x), or when P is complex number field, (x, y) = (y, x), there (y, x) the
conjugate complex number of (x, y); (2) The homogeneity: (αx, y) = α(x, y); (3)
The linearity or additivity: (x+y, z) = (x, z)+(y, z); (4) The positive definiteness:
(x, x) 0, and the necessary and sufficient conditions of (x, x) = 0 are x = 0,
then X is called the inner product space, (x, y) is called the inner product of the
vectors x and y. An inner product is also commonly denoted by the notation (·, ·). In
some books, an inner product can also be denoted by < x, y >, x · y and ( x|y) etc.
The inner product space is a special normed linear space. When the number field P
is a complex number field C, the inner product space X is called the complex inner
product space with the inner product (·, ·). When the number field P is a real number
field R, the inner product space X is called the real inner product space with the
inner product (·, ·). At this time, X is called the n-dimensional Euclidean space
about the real inner product, it is called the Euclidean space for short, it is denoted
by E
n . For example, the real space is three-dimensional Euclidean space, a plane
is two-dimensional Euclidean space, a straight line is one-dimensional Euclidean