7.3 Normal Orthogonal System and Fourier Series
401
Let X be an inner product space, x, y ∈ X , if (x, y) = 0, then x and y are called
the orthogonal, it is denoted by x⊥y. when A, B ⊆ X , for an arbitrary a ∈ A,
arbitrary b ∈ B, if a⊥b, then A and B are called the orthogonal, it is denoted by
A⊥B.
Let X be an inner product space, A ∈ X , if for arbitrary vectors a, b ∈ A (a = b),
there is (a, b) = 0, then A is called the orthogonal system; If for an arbitrary a ∈ A,
there is a = 1, then A is called the normal orthogonal system or orthogornal
system.
The normal orthogonal system can also be defined like this, let X be an inner
product space, A ∈ X , if for arbitrary vectors e i , e j ∈ A, there is (e i , e j ) = δ i j , then
A is called the normal orthogonal system or orthogornal system.
Obviously arbitrary vectors the normal orthogonal system satisfy two conditions:
One is pairwise orthogonal, another is that each vector is a unit vector, namely the
norm is 1.
The orthogonal system has the following two basic properties:
(1) For the finite abitrary vectors x 1 , x 2 , …, x n in the orthogonal system A, the
following expression holds
x 1 + x 2 + · · · + x n
2
= x 1
2
+ x 2
2
+ · · · + x n
2
(7.3.1)
As a matter of fact, since the vectors in A are pairwise orthogonal, there is
x 1 + x 2 + · · · + x n
2
=
n
i=1
x i
2
=
n
i=1
x i ,
n
i=1
x i
=
n
i, j=1
(x i , x j )
=
n
i=1
(x i , x i ) =
n
i=1
x i
2
= x 1
2
+ x 2
2
+ · · · + x n
2
(2) The orthogonal system A is a subset of the linear independence in X . In fact,
Let the vectors x 1 , x 2 , …, x n ∈ A, and
n
i=1 a i x i = 0, where a 1 , a 2 , …, a n are n
numbers, then for any 1 j n, the following expression holds
n
i=1
a i x i , x j
= a j (x j , x j ) = a j
x j
2 = 0
(7.3.2)
Since x j = 0, there must be a j = 0, x 1 , x 2 , …, x n are linear independent. Which
has proved that A is a subset of the linear independence in X .
Let X be a linear formed space, x n are a group of vectots in X , a n are a group of
numbers, there n = 1, 2, · · · , making the expression
∞
n=1
a n x n = a 1 x 1 + a 2 x 2 + · · · + a n x n + · · ·
(7.3.3)
401
Let X be an inner product space, x, y ∈ X , if (x, y) = 0, then x and y are called
the orthogonal, it is denoted by x⊥y. when A, B ⊆ X , for an arbitrary a ∈ A,
arbitrary b ∈ B, if a⊥b, then A and B are called the orthogonal, it is denoted by
A⊥B.
Let X be an inner product space, A ∈ X , if for arbitrary vectors a, b ∈ A (a = b),
there is (a, b) = 0, then A is called the orthogonal system; If for an arbitrary a ∈ A,
there is a = 1, then A is called the normal orthogonal system or orthogornal
system.
The normal orthogonal system can also be defined like this, let X be an inner
product space, A ∈ X , if for arbitrary vectors e i , e j ∈ A, there is (e i , e j ) = δ i j , then
A is called the normal orthogonal system or orthogornal system.
Obviously arbitrary vectors the normal orthogonal system satisfy two conditions:
One is pairwise orthogonal, another is that each vector is a unit vector, namely the
norm is 1.
The orthogonal system has the following two basic properties:
(1) For the finite abitrary vectors x 1 , x 2 , …, x n in the orthogonal system A, the
following expression holds
x 1 + x 2 + · · · + x n
2
= x 1
2
+ x 2
2
+ · · · + x n
2
(7.3.1)
As a matter of fact, since the vectors in A are pairwise orthogonal, there is
x 1 + x 2 + · · · + x n
2
=
n
i=1
x i
2
=
n
i=1
x i ,
n
i=1
x i
=
n
i, j=1
(x i , x j )
=
n
i=1
(x i , x i ) =
n
i=1
x i
2
= x 1
2
+ x 2
2
+ · · · + x n
2
(2) The orthogonal system A is a subset of the linear independence in X . In fact,
Let the vectors x 1 , x 2 , …, x n ∈ A, and
n
i=1 a i x i = 0, where a 1 , a 2 , …, a n are n
numbers, then for any 1 j n, the following expression holds
n
i=1
a i x i , x j
= a j (x j , x j ) = a j
x j
2 = 0
(7.3.2)
Since x j = 0, there must be a j = 0, x 1 , x 2 , …, x n are linear independent. Which
has proved that A is a subset of the linear independence in X .
Let X be a linear formed space, x n are a group of vectots in X , a n are a group of
numbers, there n = 1, 2, · · · , making the expression
∞
n=1
a n x n = a 1 x 1 + a 2 x 2 + · · · + a n x n + · · ·
(7.3.3)