402
7 Variational Principles
Equation (7.3.3) is called the series in X . If taking the former n terms
S n =
n
k=1
a k x k
(7.3.4)
then S n is called the partial sum of the former n terms for the series. If the number
sequence of partial sums {S n } converges to S ∈ X , then the series
∞
i=1 a i x i is called
the converges to sum S, it is denoted by
S =
∞
n=1
a n x n = lim
n→∞
n
k=1
a k x k
(7.3.5)
If the series
∞
n=1 a n x n =
∞
n=1 |a n ||x n converges, then the series (7.3.3) is
called the absolute convergence or absolutely convergent.
Lemma 7.3.1 Let A be the normal orthogonal system in an inner product space X ,
taking n arbitrary vectors e 1 , e 2 , …, e n in A, α 1 , α 2 , …, α n are n arbitrary numbers,
then the following inequality holds
x −
n
i=1
(x, e i )e i
2
= x
2
−
n
i=1
|(x, e i )|
2
0
(7.3.6)
x −
n
i=1
α i e i
x −
n
i=1
(x, e i )e i
(7.3.7)
Proof According to the definition of norm, there is
x −
n
i=1
α i e i
2
=
x −
n
i=1
α i e i , x −
n
i=1
α i e i
= (x, x) −
x,
n
i=1
α i e i
−
n
i=1
α i e i , x
+
n
i=1
α i e i ,
n
i=1
α i e i
= (x, x) −
n
i=1
α i (x, e i ) −
n
i=1
α i (e i , x) +
n
i=1
α i e i ,
n
i=1
α i e i
= (x, x) −
n
i=1
α i (x, e i ) −
n
i=1
α i (x, e i ) +
n
i=1
α i α i (e i , e i )
= (x, x) −
n
i=1
α i (x, e i ) −
n
i=1
α i (x, e i ) +
n
i=1
|α i |
2
0 (7.3.8)
Putting α i = (x, e i ), then there is