1549301742-The_Theory_of_Difference_Schemes__Samarskii

134 Basic Concepts of the Theory of Difference Schemes

To make sure of it, we take the inner product of (37) and y = A-^1 <p:

or II y II~ = II <p II~_,.

3) If A > 0, then

( 41) A 0 = ~ ( A + A* ).

Furthermore, taking the inner product of (37) and y we obtain

( 42) (Ay,y)=(<p,y),

where A 0 = ~ (A+ A*) is the syrnmetric part and A 0 = ~ (A-A*) is

the skew-symmetric part of the operator A. This provides the sufficient

background for the relation ( A y, y) = ( A 0 y, y) due to the fact that

( A 1 y, y) = 0. As far as A 0 > 0, the inverse operator A;;-^1 exists, be-

cause the space H is finite-dirnensional *) and, therefore, by the generalized

Cauchy-Bunyakovskil inequality one can write down

Substitution of this inequality into ( 42) leads to the relations

from which the desired estimate ( 41) immediately follows.
4) If A > b E with b > 0, then

( 43)

To prove this fact, we make use of estimate (41) and the inequality

which is a consequence of

A 0 > b E and

*)If H is an infinite-dimensional space, one should require instead of A > 0 that
A ?: 5 E with 5 > 0.