33 Operators in Normed Linear Spaces

33 I

Operators in Normed Linear Spaces

MAPPINGS AND OPERATORS 1. Consider two NLS Xand Y, where it could be also Y = X , and a subset,

U,of X.If we want to define a “function” f with the “argument” running

through U and the “values” from a subset, V , of Y, we have t o interpret it as a mapping of U into V by which to any element 5 E U corresponds its image, an element q of V . Here “into” means that the images of the elements of U do not necessarily cover the whole V-otherwise we say “onto” instead of “into.” In this connection it is usual in functional analysis to speak, not of a function in U,but of an operator acting on U. Further, instead of f ( 5 ) = q we write f 5 = q, which then means that q is the image of 5 by the mapping f, that is, thatftransforms 5 into q. Instead of the locution the operator fmapping U into V we will simply write f(U--t V ) . The operator which, applied to the elements of A’, maps each element into itself is called the identical operator and is usually denoted by I. If g(U+ Y ) is another operator of the same kind, we define for any a, b from C, and for any 5 from U: (af+W

5

:= a V 5 )

+ b(g5).

(33.1)

We have here obviously a new operator mapping U into Y which is denoted by af + bg. Tn the sense of the addition and multiplication operations defined in this way, the set of all operators mapping U into Y is a linear space. The zero element of this space is the zero operator which maps all elements of U into the zero element of Y . BO UN D E D 0P ERATO RS

2. The operator f(U Y ) is called bounded on U if there exists a B 2 0 such that

(33.2)

220

33

I

OPERATORS IN NORMED LINEAR SPACES

In (33.2), of course, the left- and right-side norms are taken respectively in Y and X . Iffis a bounded operator, denote by Bf*the Infimum of all Bin (33.2). Then (33.2) remains valid if B is replaced by Bf*,as follows at once by going conveniently to the limit. The set S u ,y,of all bounded operators mapping U into Y is an NLS if we put

llfll

:=

Bf*

SV,d.

(33.3)

Indeed,wehave,ifaczCy,b~Cy, fESU,Y,gESU,Y,

II(d+bs)5II

llaf511+ llb95II G l~lBf*11511+1~1~g*l1511

+lbl47*)Il5ll.

(lalB,*

We see that af+ bg is also a bounded operator and it follows further that B:f+bg

G

I4 Bf*+ Ibl Bg*.

(33.4)

For a = b = 1 we have the triangle inequality. For b = 0, a # 0 it follows, using thatf= (l/a)(af),that

which gives the homogeneity relation. Further, if Bf*= 0, it follows for all 5 E u: Ilf’tII = 0,ft= O,f= 0. The norm off defined as above is called induced by the norms in X and in Y. The simplest bounded operator mapping X into X is q = a t , realized by multiplying any element 5 of X with a Gxed a E C,. If X = R,y = sinx is a bounded operator (R + R), if R is normed by the moduli; the induced norm of sin x is 11 sin xi1 = 1. On the other hand, the operator (R -+ R) given by y = x 3 is obviously not a bounded operator if Iw is normed by the moduli of its elements. LINEAR OPERATORS

3. An operator ( X + Y ) is called additive if we have ( 5 1 E x,5 2 E X I . f(tl+52) = f51 + f 5 2 An operator ( X -,Y ) which is both bounded and additive is called linear. The

simplest linear operator ( X + X ) is realized by a t where the “constant” a belongs to C,. If X = R, this is also the most general linear operator. This is, however, no longer necessarily true if X is, for instance, @. In this case we can define a linear operator in the following way: Consider arbitrary nonnegative

STRONG A N D WEAK CONVERGENCE E,

6 ; put, for 5 = a+ bi,

ft

:=

a&

22 1

+ b6 = R ( E - i d ) t .

It is easily seen that this formula defines a linear operator. However, this can only be expressed in the formft = (a ip) 5 for fixed real a and p, if we have 6=&=0.

+

4. Observe finally that if L is a linear operator in X , we have, for 5 E X , CEX, L(C+C)- L(5) = U5), IIU5+5) - UOII G IlLll 11511.

We see that such an operator satisfies in the whole space X a “Lipschitz condition” with the “Lipschitz constant,” IILIJ. It follows in particular that L is a “continuous function” of its argument, 5. On the other hand, it followsfrom the additivity immediately that for any integerp, L(p5) = p L ( < ) ,and therefore, for any rational r, L ( r 5 ) = rL(5).

If now a is an irrational real number, we have for a sequence of rational real numbers rvrconvergent to a, L(rv 5 ) = r, L(5) and therefore in the limit, using the continuity property of L,

W 5 ) = aU5)

(a 5-0).

(33.5)

STRONG AND WEAK CONVERGENCE

5. Assume that we have in Su,

llfv-fll

f, +f,

+

0.

(33.6)

This is called the strong convergence in Sv,y. From (33.6) it follows for any t e Uthat l l f v 5 - f 5 I I G llf,-fIl11511-0~ f v 5 +f5 (5 E U), (33.7) and even the uniform convergence: fv5

*f5

(t E u, 11511 G C ) .

(33.8)

On the other hand, if we have, for f E Su,y, f v E Su,y, the relation (33.7), we say that the sequence f,converges weakly to$ While, as we have seen, the weak convergence follows from the strong convergence, the converse is not necessarily true. However, we can prove the Lemma. If the sequence fv-fromSu9 is a Cauchy sequence and is weakly convergent to a n j E Su,y , f v also strongly converges to$

33

222

I

OPERATORS IN NORMED LINEAR SPACES

6. Proof. Put g,

Then we have

:=fV-J

119v-9p11

-+

(Min(w)

0

gpt40

(5E

+

a),

(33.9) (33.10)

U)

and have to prove that gv + 0, that is, ]]gvll -,0. Since lim llgvII exists by Section 4 of Chapter 31, we will assume that our assertion is false and therefore that I19rII

+

P

'0.

(33.1 1)

By the definition of the norm there follows from (33.11) the existence of a sequence of nonzero elements from U such that

<,

Since further by (33.9)

"gp"" -+ p

I1t v I1

(Min(p, v)

+

co).

But then there exists an integer N such that

and therefore in particular taking v = N IlgfirNJI

P > j IltNll

012 N )

and this contradicts (33.10). Our lemma is proved. 7. As a corollary of the lemma in Section 5 we prove now Theorem 33.1. I f Y is complete, then Su, is also complete.

Indeed, consider a Cauchy sequencef,: ~ ~ f v -+fOp ~ (Min(v,p)-, ~ for any fixed

<

llfvt-fptll

IItll

00);

then

0 and thereforef, 4 is a Cauchy sequence. From the completeness of Y it follows thatf, tends to an q contained in Y. In this way we have a mapping of 5 on q

<

llfv-&ll

+

STRONG AND WEAK CONVERGENCE

which gives us an operator f ( U + Y ) with q Section 4 of Chapter 31, w := lim v-m

=f 5 .

223

Put, using the remark in

]]h]l;

then it follows that and we see thatf is a bounded operator and belongs therefore to Su,y. But then f , is weakly convergent to f and our assertion follows from the lemma in Section 5. 8. Consider two n-dimensional vector spaces C": X ( 5 1= (xl, ..., x,,)), Y ( q := (yl, ...,y,,)). If f ( X + Y ) is a linear operator, it can easily be expressed by a matrix. Indeed, denote the coordinate unity vectors by 5'') := (61v,62,, ..., 6,,,) (v = 1, ..., n), where 6,, is Kronecker's 6. Then 5 = C : = , X , P. Put now f t " ) = (a,,, ..., unv)and consider the matrix A := (up,). Then in the relation q =ft the components of q are given by y, =

2 apvxv

v=l

(p = 1,

..a,

n)

and this is the matrix formula q' = At'

which describes the most general linear operator mapping X into Y .

(33.12)