400
7 Variational Principles
Equations (7.2.19) and (7.2.20) are usually called the polarization identity.
Proof When X is a real inner product space, imitating the proof of the Lemma 7.2.2,
we have
x + y
2
− x − y
2
= (x + y, x + y) − (x − y, x − y)
= (x, x + y) + (y, x + y) − (x, x − y) + (y, x − y)
= (x, x) + (x, y) + (y, x) + (y, y) − (x, x)
+ (x, y) + (y, x) − (y, y)
= 2(x, y) + 2(y, x) = 4(x, y)
Equation (7.2.19) is proved.
When X is a complex inner product space, by the above calculation, we get
x + y
2
− x − y
2
= 2(x, y) + 2(y, x)
Changing y in the above equation into iy, then multiplied by i, we get
i(x + iy
2
− x − iy
2
) = i[2(x, iy) + 2(iy, x)]
= 2ii(x, y) + 2ii(y, x) = 2i(−i)(x, y) − 2(y, x)
= 2(x, y) − 2(y, x)
Adding the above two equations, we obtain
x + y
2
− x − y
2
+ i(x + iy
2
− x − iy
2
) = 4(x, y)
Equation (7.2.20) is proved. Quod erat demonstrandum.
The continuous part in the n-dimensional Euclidean space E
n is called the domain
or region, the border of the domain is denoted by Γ . In the domain, the part that
does not include the border Γ is called the open domain, it is denoted by G or D
etc.; The domain including the bonder Γ is called the closed domain, it is denoted
by G or D etc.
7.3 Normal Orthogonal System and Fourier Series
Let x ∈ X , if there exist the numbers α 1 , α 2 , …, α n ∈ P, x 1 , x 2 , …, x n ∈ X , such
that x =
n
i=1 α i x i , then x is called the linear combination of x 1 , x 2 , …, x n .
Let x 1 , x 2 , …, x n ∈ X , There exist n not all zero numbers α 1 , α 2 , …, α n , such
that
n
i=1 α i x i = 0, then x 1 , x 2 , …, x n are called the linear dependence, linear
correlation or linearly dependent, or else they are called the linear independence
or linearly independent. In other words, if
n
i=1 α i x i = 0 holds, there must be
α 1 = α 2 = · · · = α n = 0, then x 1 , x 2 , …, x n are the linear independence.