1549301742-The_Theory_of_Difference_Schemes__Samarskii

42 Basic Concepts of the Theory of Difference Schemes

approved rule governing what can happen a vector y = Ax E Y, we say

that an operator A is given on D (or in X) with values in Y. The set D is

called the do1nain of the operator A and is denoted by D( A). The set of all

vectors of the form y = Ax, x E D( A), is called the range of the operator

A and is denoted by R( A). It is customary to use also the well-established

notation A( x) instead of Ax.

Two operators A and B are said to be equal if their domains coincide

and for all x E D(A) = D(B) the condition Ax = Bx holds true.

An operator A is called linear if it is:

1) additive, meaning that for all x 1 , x 2 E D(A)

2) ho1nogeneous, meaning that for all x E D(A) and any number ,.\

A(..\x) =..\Ax.

A linear operator A is said to be bounded if there is a constant M > 0

such that for any x E D(A)

(1)

(here II · 111 and II · 112 are ad1nissible forms of the functional norms on the

spaces X and Y, respectively).

The 1ninirnal constant NJ satisfying condition ( 1) is called the norm

of the operator A and is denoted by l!Allx~Y or simply II A II·

(2)

It follows from the definition of norm that

!!All= sup !!Ax!! 2

11x111 =1

or !!A!!=sup l!Axll2

x,to II X (^111)
It is worth noting here that in a finite-dimensional space any linear operator
is bounded. All of the linear bounded operators from .X. into Y constitute
what is called a normed vector space, since the norm II A II of an operator
A satisfies all of the axioms of the norm:

1. II A II> O; if II A 11 = 0, then II Ax 112 = 0 for all x and A= O;


2. II.AA!!= I.A!· !IA!!;


3. !!A+B!! <!IA!!+ !!Bil·
   We will denote by X f-7 ){ the set of linear bounded operators with the
   domain coinciding with X and the range belonging to X. On the set X ---> X