1549301742-The_Theory_of_Difference_Schemes__Samarskii

46 Basic Concepts of the Theory of Difference Sche1nes

(x, y)A = (y, x)A or (Ax, y) = (x, Ay) = (Ay, x). The axioms of the inner

product imply the Cauchy-Bunyakovski1 inequality

and the triangle inequality

This profound result is covered by the following assertion.

Lemma 2 For any positive self-adjoint operator A in a real Hilbert space

the generalized Cauchy-Bunyakovskil inequality holds:

(9) (Ax, y)^2 <(Ax, x) (Ay, y).

Remark The preceding inequality remains valid in the case when A is a

nonnegative operator.

If A is a self-adjoint operator for which A-^1 exists, its "negative" norm

can be defined by

(10)

In this line, we claim that

( 1 O')

l(<p,x)I

II <p llA-1 = sup x;tO II J: II A

Indeed, we deduce from inequality (9) that

yielding

sup

x;to

On the other hand, for x = A -l <p it is plain to show that

(<p,A-1<p)

j(AA-l<p,A-l<p) = ll<pllA-1,