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7 Variational Principles
product. Equations (7.2.1)–(7.2.3) are all the concrete manifestations of Eq. (7.2.4).
Equation (7.2.5) is called the Dirichlet inner product.
Theorem 7.2.2 Let X be a inner product space, if for arbitrary x, y ∈ X , then there
is the following inequality
|(x, y)|
2
(x, x)(y, y)
(7.2.14)
Inequality (7.2.14) is called the Schwarz inequality, it was obtained by Schwarz
in 1885. The Schwarz inequality is sometimes also called the Cauchy-Schwarz
inequality.
Proof If x = 0 or y = 0, then the equal sign in the above inequality holds, if y = 0,
then for an arbitrary α ∈ P, from the property of the inner product, there is
0 (x + αy, x + αy) = (x, x + αy) + (αy, x + αy)
= (x, x) + ¯
α(x, y) + α(y, x) + α ¯
α(y, y)
According to the complex function theory, the product of the two conjugate
complex numbers is equal to modulus square of one of the two, namely
zz = (x + iy)(x − iy) = x
2
+ y
2
= |z|
2
= |z|
2
Putting α = −
(x,y)
(y,y)
, instituting it into expansion of the inner product (x + αy, x +
αy), we obtain
(x, x) + α(x, y) + α(y, x) + αα(y, y)
= (x, x) + α(x, y) −
(x, y)
(y, y)
(y, x) −
(x, y)
(y, y)
α(y, y)
= (x, x) −
(x, y)
(y, y)
(x, y) = (x, x) −
|(x, y)|
2
(y, y)
0
Thus the Schwarz inequality (7.2.7) is obtained. Quod erat demonstrandum.
Theorem 7.2.3 Let X be a inner product space, x ∈ X , putting
x =
(x, x)
(7.2.15)
then x is the norm of X , it is called the norm generated by the inner product or
norm induced by the inner product, under the norm, X becomes a normed linear
space. The norm induced by Eq. (7.2.7) is sometimes called the energy norm.
Proof It need only to certify that the norm of the above definition satisfies the three
axioms of the norm. The positive definiteness and homogeneity of axioms of the
norm are clearly valid, in fact