Issuu

JOSE SEBASTIAO E SILVA

TEXTOS DIDAcTICOS

Volume III

SERVI<;O DE EDUCA<;AO E BOLSAS

FUNDA<;AO CALOUSTE GULBENKIAN

LISBOA

Reservados todos os direitos de acordo corn a lei Edic;ao da FVNDA<;AO CALOUSTE GULBENKIAN Av. de Berna - Lisboa

1999 ISBN 972-31-0971-9 Dep6sito Legal

n.O

148805100

111.1 THEORY OF DISTRIBUTIONS路

Este texto tern por base apontamentos coligidos por diversos alunos de Jose SebastHio e

Silva na sequencia de urn curso que realizou em

1958 na Universidade de Maryland, e que

posteriormente foram utilizados, e por ele revistos, na Faculdade de Ciencias de Lisboa.

C H A PTE R V TO PO LOG I ES O N S PAC ES OF D I STR I B UTI O N S

5.1 . Limit of a sequence of continuous functions

Let I be a compact interval in fR. Then, if f is any continuous function on I, the supremum of I f(x) I on I is a non-negative real number, called the norm of fE C(/) and denoted by I l f ll .

I l f ll = sup I f(x) I = max I f(x) l · xEI

xEI

With this definition, the vector space space.

C(/) becomes a normed

5. 1 . 1. A sequence fo ' . . . ' fn , of vectors of C(/) converges in norm to a vector g of C(/) iff Il fn- g ll � O as n � oo . Then we write fn � g in norm on I. •

•

It is well-known that this notion of convergence is equivalent to that of "uniform convergence" on I. Remember that the sequence fn is said to converge uniformly to g on I iff for every 0> 0, there exists an integer N (depending on 8, but not on x), such that:

I fn (x) - g (x) l < o , for all n > N and all xE /.

Remember also, that uniform convergence is a sufficient con dition for the limit operation to be interchangeable with the integral operator.

5.1.2. If the sequence of continuous functions fn converges uniformly on I to a (continuous) function g and if c is any point of J, then the

sequence of (continuous) functions Fn (x) =

ffn(; ) d;

(n = 1 , 2 , . . . ; x E I )

converges uniformly on I to the function G(x) = Xg (;) d; . To see that, it is sufficient to apply the hypothesis observing that:

I I;; (x) -G(x) l s

J:i fn(; )-g(; ) i d;s I II �::f l fn(; )-g(; ) 1 s 1 11 1 1 fn-g ll< I I I I � III I l fn- g l l < � I :Jf(x) = f ; ;

if n is such that

, where

Thus, if we put press

denotes the lenght of /.

f( ) d , for any fE C(I), we can ex

5 . 1 .2. by the formula:

5.1.3. lim ( � fJ = � (lim fn ) whenever fn � g in norm (or by saying: '

the operator � is continuous on the normed space C(l)). On the contrary, the operator D is not continuous on the sub space C 1 (J) of C(J), with the same norm. For example, consider

fn (x) = 1 sin (nx) , for n = l , 2 , . . . , then sup l fn (x) 1 = 1 on IR for any n, n n

fn converges to 0 uniformly on every compact subinterval I (even on IR) ; but the sequence of derivatives f� (x) = cos nx does not so that

converge uniformly (or even point-wise) on any compact interval !.

The s pace !fif(/) was constructed (2.2.) in order to make the n operation D always feasible, on continuous functions . Our next purpose is to define a suitable topology on !fif(l) so as to render the

same operation continuous.

For that purpose, we begin with the concept of convergence of distributions . "

Notation. In all subj ects about limits, we shall use the symbol � as an abbreviation of "converges to", "tends to" or "approaches".

5.2. Limits of sequences of distributions

Let us consider first a compact interval I on fR. The concept of convergence for sequences of vectors in the space !fif(l), is defined so as to guarantee at least the two following properties :

If a sequence of elements io ' ' in ' . . . E !fif(/) converges to an element g of !fif(l) then the sequence of derivatives Dio ' · · ' Din ' . . . converges to Dg; that is: L1.

•

in � g imp lies Din � Dg .

If a sequence offunctions in E C(/) converges uniformly on I to a function g E C(/) then in � g in the distributional sense (i. e. ac cording to the new concept which we will define). L2.

From L 1 and L2 it follows immediately that if a sequence of functions inE C(/) converges uniformly on I to g then D Pin � D Pg , for any integer p.

DEFINITION. We say that a sequence of distributions in on I converges (or tends) to g E !fif(l) iff there are a fixed integer p, a se quence of functions Fn E C(l) and a function G E C(/), such that: 5.2. 1 .

f.n= DPFn ' for all n ,' (ii) g = DPG ;

(i)

(iii) Fn converges uniformly on I to G. It is obvious that this definition satisfies L 1 and L2 . Therefore, we shall see that: 5.2.2 · lf in � g and in � g *, then g = g *. In fact suppose : 3p, q E INo ' sequences Fn , Fn* E C(l) and G, G* E C(l) such that:

G) f.n= DPFn= D qFn* ' for all n', Gj) g = DPG and g * = D qG * ; Gjj) Fn and Fn* converge uniformly on l, to G and Assume for example p � q and set p C"A -qFn* .j: lor Pn- Fn - �

G* respectively.

n - 0, 1 , . . . .

Then by G), every Pn E oCfPp and according to 5 . 1 .2. � d Gjj) Pn converges uniformly on I to the function Q = G - C;S p - qG* . Hence Q E CfPp and according to Gj), g g *. • =

Remark. We have here used the well-known property : "If a se quence of polynomial functions 1!n of degree < r is convergent at, at least r distinct points Xl ' " ' ' xr ' then 1!n converges to a polynomial function 1! of degree < r at every point X (even uniformly on every compact interval)" . This property can be proved with the aid of the Lagrange interpolation formula; remember that: r

�l ({J/x) 1!n(xk ), where

1!n (x) ({Jk (

(X-X 1 ) . . · (x-xk-l )(x-xH I ) . . . (x-x )

(X -X ) . . · (x -X k

k- l

) (Xk-XH I ) . . · (Xk-Xr) r

Then by putting

li,: nn(x)

ck Um n nn(xk ) for k= I , . . . , =

�1 c/Pk(x) x n(x) �1 C/Pk(X), r

pol ynomial

for every

and this shows that

we have

E fR, and therefore the limit is the

of degree < r. On the other hand, if we

put for every compact interval

I n/x) - n(x) I

I, M(I) = maxi lPk(x) I in I, we find: k x,

sk� M(/) I nn(xk ) - ck l , 'tIx E / , n = r

nn � n uniformly on I.

This property can be expressed by saying: for set CZPr is closed in the normed space C(/).

1 , 2, . . .

all integer r, the

5.2.3. DEFINITION. If fn � g in !?fl(/), we say that g is the limit of the sequence fn and we write g = limfn · Observe that the uniqueness of the limit is assured by 5 . 2 . 2 . , which in turn is a direct consequence of definition 5 .2. 1 . Now it is readily seen that:

5.2.4. If fn� f * an d gn � g * I n �(/) and if a, f3 E e, then

fn + f3gn � af* + f3g *; hence lim ( afn + f3gn ) = a lim fn + f3lim gn · More generally, this is also true if a and f3 are C oo functions on I. a

Let us now consider any

open set Q in fR .

5.2.5. DEFINITION. We say that a sequence fn in �(Q) converges to g in �(Q) iff for every compact interval I E Q, the sequence of distributions P[fn converges to PI g in �(/) according to definition 5 .2. 1 . It is a simple matter to extend properties L l , L2, 5 .2.2. and 5 .2.4. to this new concept. For 5 .2.2. apply 2. 8 . 5 . The uniqueness property enables us to write g = lim fn iff fn � g also in this case.

Observe that definition 5 . 2 . 5 . extends immediately with the same properties, to the space !!lJ (.Q) of global distributions on .Q (cf. 2 . 8 ) . For example, we can represent by

the global distribution considered in 2 . 8 .4.

5.3. Convergence in the mean and convergence in distributional sense. Examples. Let us consider again a compact interval I. In the vector space L (/), of all summable functions on I, the norm 11 f i l l of a vector f is usually defined by :

A sequence of vectors fn in L (I) is said to converge in the mean to a vector g in L (/) iff 11 fn- g I I I

� o.

5.3.1. If a sequence offunctions fn in L (/) converges almost every where (a. e.) in I to g and if there exists a number M such that I fn (x) l s M for all n = O, 1 , . . . and all x in I, then g E L (/) and in con

verges to g in the mean.

This is an immediate consequence of the Lebesgue theorem.

5.3.2. If in converges in the mean to g, then fn converges to g in dis

tributional sense.

PROOF. Suppose that Jlf I fn- g l

G(x)

fg(g) dg ,

where

�o

c E l. Since

and put Fn (x)

JcfXfn (g) dg,

it is seen that Fn � G uniformly . on I . As follows that fn � g in distributional sense •

fn= DFn and g =D G , it

Example. Consider the sequence of functions:

fPn (x) =

Then, if we put have

cPn (X) =

n ' ( n = O, 1 , . . . ). + 1 (nx) 2

lPn (x) = Jofx qJ,n (g) dg for n = O, 1 , . . . and all x in IR, we

( �) arctan (nx)

and hence:

I lPn (x) 1 :S "21 ' for n = O, 1 , . . and any x E IR, lPn (x) = nlim -- oo

{

112 , if x > O 0 , if x = O -1/2 , if x < O.

0.5

-0.5

lP/x) converges at every point x of IR, except zero, to

the function

H(x) -

�. Therefore,

according to 5 . 3 . 1 . and 5 . 3 . 2 . ,

cP,

converges in the distributional sense to H(x) l on every compact 2 interval in IR, and hence on IR, according to definition 5 . 2.5 . But _

£Pn = D cPn for every n. Hence £P

•

....,.

."..

---

- ::.. - ::. - ;. .=. ; -... --

/- .::. - -

- - -- -

(

8, that is 8 = lim £P

• .

- - ... - - ..

- -

- ... .. ... .. ....

...... ..... -

-:,

.... -- ... -:..:

-:: -=-.- :.: :... -

- - - - - - - - - - -- - - .

More generally, since ({In is a C function on IR for each n, we have: Cl:)

8 (k) = lim ({J�k\ for n = O, 1 , . . .

({In

�

8 � ({J: � 8 ' � ({J;' � 8 " � . . . � ((J�k)� 8(k).

So the distribution 8(k) is expressed as the limit of a sequence of C ) ({Jo(k , . . . , ({In(k) , . . fiunctlons, · .

Cl:)

•

5.4. Inductive limits ; (LN*)-spaces Till now, we have only defined a concept of convergence for sequences of distributions. In order to define in the preceding spaces of distributions a suitable topology leading to that concept of convergence (which is however sufficient for the following chapters) we need some special notions and results concerning the theory of locally convex spaces.

C on sid er a vector space E, a family (E)a EA of vector spaces over the same field (lR or C ) and let fPa be, for every a EA, a linear mapping of Ea into E. Suppose that, on each space Ea is defined a local ly convex topology 'fa (not necessarily Hausdorff) . Then it is easily seen that among all locally convex topologies for which the mappings fPa are continuous for all a , there exists one 'f * which is stronger than all the others . (A fundamental system of neighbor hoods of 0 for 'f * may be the family of all circled convex and ab sorbing subsets "D of E such that for every a , fP� ("D ) is a neighbor hood of 0 for 'fa ) . That being so, 'f * is said to be the inductive limit of the topologies 'fa · In particular, E may be the union of all Ea and fPa the inj ection (identity mapping) Ea-::. E. Then E( 'f * ) is said to be the inductive limit of the spaces Ea( 'fa ) . In this particular case 'f * is the strongest locally convex topology on E, inducing on each Ea ' a topology weaker than 'fa .

5.4.1. DEFINITION. A locally convex space E is called a (LN*) space iff E can be represented as the inductive limit of a sequence El ' . . . ' En ' · · · of normed spaces such that: ( 1 ) En C En+l ' for all n, (2) the injection En -::. En+1 is, for all n, compact (which means that all bounded sets in En are relatively compact with respect to the nonn of En+1 ) . Such a sequence En is said to be regular. 5.4.2.

LEMMA. For every regular sequence (E) or normed sp aces,

there exists an increasing sequence (FJ of Banach spaces such that: ( i) every bounded closed set in Fn is compact in Fn+ 1 ; (ii) fior all n, En C Fn C En+1 and the iniections En -::. Fn , Fn -::. En+ l ' are '.J continuous. PROOF. Let E be the inductive limit of (E). For every n, denote by

1 1 · ll n the norm in En and set Bn = { x : I l x ll n < 1 } ;

i(= closure of Bn in En+1 ; En= U kEn . k= l

As the injection En ----;;. En+l is continuous for all n, we can suppose that the norms 11 · ll n have been chosen so that Bn CBn+ 1 ; i . e · l l x l l n � I l x l l n+ l for all n. Then Bn C Bn C Bn+ 1 ' Since Bn is circled and convex so is Bn , and ----

therefore En is a vector subspace of En+1 • Besides, if we place ----

gn (x) = inf{p > O : x E pBn } , Vx E En , gn will be a norm defined on En (since Bn C Bn+1) . Thus Fn= En be comes a normed space and (ii) follows immediately from the double inclusion BnC Bn C Bn+ I for all n. Moreover, every bounded closed set H in Fn will be compact in Fn+l' since there exists p>O such that

H CpBn and Bn is compact in En+ l hence in Fn+l (which induces in En+l a weaker topology). It can be also proved · that the spaces Fn are complete, but that is not required for the following applications . • Observe that according to condition (ii), the sequences

(FJ have the same inductive limit.

(EJ and

In the following propositions, (EJ denotes a regular sequence, E the inductive limit of (En )' hence a (LN*) space; 1 1 · 1 1 n denotes the norm of En ' Bn = {x : I l x l l < I } . 'l'n is the topology on En given by 1 1 · l l n and 'l'oo the topology of E (inductive limit of the topologies 'l'n ) We '

can assume without loss of generality that Bn C Bn+ 1 for all n and that

Bn is compact in En+l (according to the lemma). That being so

5.4.3. THEOREM. A set H is closed in E ifffor every n, H n En is 'l'n closed. PROOF. a) Suppose H is closed in E . Since 'l'n induces in each

En a topology weaker than 'l'oo ' H n En must be 'l'n -closed. b) Suppose Hn En is closed in En for all n and H is non-empty (if H= 0, the statement is obvious) . Let Xo be any point of E such that

Xo f!:. H. We must prove that there exists a 'l'oo neighborhood of Xo whose

intersection with H is empty. Since xo E E=

U En

and H � 0, there

ex i sts at least one p such that xo EEp and H n E � 0 . We may assume p with out loss of generality, that p = 1 . Now it suffices to show that n

there exists an increasing sequence of circled, convex sets U I ' U2 , such that for all

•

(i) Un i s neighborhood of 0 i n En ' (ii) Xo + Un does not intersect H.

In fact, if such a sequence ( Up ) exists, then Xo+

xo -neighborhood of

U Un 00

will be a

0 (by the definition of inductive limit) which

does not intersect H by virtue of (ii) .

Since H n E I is closed in El ' and xoE E I we can choose a ball H I of center 0 in El ' such that xo + UI does not intersect H n E I • Suppose

now that we have already chosen

n sets U I , .

, Un ' satisfying the

preceding conditions. Then Xo + Un is compact in En +l , and .

does not

intersect H. On the other hand, H n En +1 is closed in En +l . Therefore

the distance � between xo + Un and H n En + l , in the normed space En + 1

{

must be > 0 . We set �+ I = x : I l x l l n+ l <

;}

and Un+ 1 the circled convex

hull of Un U '-':+1 ' Then Un C Un +1 C Un U '-':+1 ' so that Xo + Un+1 cannot in

tersect H, the distance between Un U Vn + I and H n En + I being

�

En+l · On the other hand, since Un and '-':+ 1 are bounded in En+ l , so is

their circled convex hull, that is Un+ l• Finally, as '-':+1 is a neighbor

hood of 0 in En+l , so is Un+ 1 � '-':+1 ' Thus all the preceding conditions are satisfied by the sets UI , , Un+1 and the theorem is proved by

induction on n . •

5.4.4. COROLLARY.

•

Every (LN*) space is a Hausdorff space.

In fact, the theorem implies that every set reducing to a point is

roo -closed.

5.4.5. COROLLARY. Let F be any topological space. Then a map ping qJ : E � F is reo continuous if! its restriction to each En is a 'fn -con tinuous mapping of En into F. PROOF. Suppose qJ is 'feo continuous and let M be any closed subset of F. Then qJ-I (M) is reo closed and hence qJ- I (M) n En is rn closed. The converse is analogously proved . •

Remark. This corollary is equivalent to the theorem itself. What the theorem means is that among all topologies in E, (not necessarily locally convex), inducing on each En a topology weaker than rn ' reo is the strongest one. 5.4.6. THEOREM. A set H is bounded in E if! there exists an integer p, such that H is contained in Ep and bounded in this normed space. PROOF. a) Suppose H is contained and bounded in Ep ' and let V be any 'feo neighborhood of 0 in E. Then V n Ep contains a � -neigh borhood of 0; i.e. there exists an t:> 0 such that t:Bp C V n Ep ' On the other hand, H being bounded in Ep implies that there exists a p > 0 such that H C p (t:Bp ) ' Hence HC p V; i.e. H is absorbed by any reo -neighborhood V of 0 and therefore is bounded in E. b) Suppose now H is bounded in E, and put

Cn = { x : Il x ll n < n } = nBn .

We are going to show that H is contained in one of the open balls Cn Suppose this is not true. Then it will be possible to take in H a sequence of points Xl " ' " xn " " such that xnft. Cn for all n. Now by a technique similar to the one used for theorem 5 .4 . 3 . we are going to prove the existence of a sequence UI C U2 C . . of circled convex sets such that: (i) Uk is a bounded and closed neighborhood 1 of 0 in Ek contained in Bk for all k ; (ii) - xn ft. Uk for all k and n. For •

example, take then

� = 2, B I ; since xn ft. Cn= n Bn and B I C Bn for all n, 1

xn ft. UI for all n. Suppose that we have already chosen k sets

� . . . , Uk satisfying the preceding conditions and place: ,

Then Mk is a closed set in Ek+1 which contains all points

xn and

does not intersect Uk , according to (i) and (ii). On the other hand, Uk is

{

compact in Eh1 . Hence, if we set q = dist ( U" M, ) , v, = x : I l x 11 , <

�}

and �+1 = circled convex hull of Uk U Vk ' we can prove, as in theorem 5.4. 3 . , that �+1 is a circled convex hull compact in Ek+2 contained 1 in Bk+1 , and it is obvious that - xn � �+1 for all n. Thus the existence o

of a sequence satisfying (i) and

(H) is now proved.

Now it is readily seen that the set V

U Un is a 'oo-neighborhood 1

1 of ° such that - xn � V. But then, for every p > O, we should have

n xn � p U for all n > p and this is impossible, the set H being bounded in E. Consequently, there exists at least one p such that He Cp , which implies that H is contained and bounded in Ep . • 5.4.7.

COROLLARY. A sequence (xn ) of points of E converges to a

point x of E if and only if there exists a p such that all the points xn and x belong to Ep and xn � x in Ep . PROOF. a) Suppose there exists p such that xn � x in Ep . Then, since ' induces on Ep a topology weaker than -z;" xn� x in E. b) Suppose xn� x in E. Then the set X {xn } n EIN U {x} is bounded in E and so there exists an r such that X is bounded in Er . Hence the aderence X of X in Er+l is compact and, according to a general theo rem of Topology, ' induces on X the same topology as does 'r+ . So, l as xn � x in E, we can conclude that xn � x in Er + 1 • • (x)

(x)

5.4.8. COROLLARY. If E

is infinite dimensional, then E is not

metrisable. PROOF. Suppose

mental system 't) =

't)

E is metrisable . Then there exists a funda of neighborhoods of 0 in E which is countable :

{� , . . ., �, . . . }.

Now at least one of these neighborhoods mu st

be bounded in E; otherwise there would exist, for each o

n, an xn E �l

xn tl. Cn= nB (by the theorem) and thus (xn ) would be a sequence converging to 0 and unbounded, which is impossible. Let � be a neighborhood of the system 't) which is bounded in E, then � is bounded in Em for some m and hence relatively compact in Em + l • As � is also a neighborhood of 0 in Em+1 ' it follows that Em+1 is finite di . = E since Vp is a neighborhood of 0 in mensional and Em + Em +2 such that

E which implies E = U k � . • 00

Every (LN*) space E is an (M)-space (i. e. a Montel space where every bounded set is relatively compact) . In fact if M is a bounded set in E, then M is bounded in some 5.4.9. COROLLARY.

Ep and hence relatively compact in Ep + I ' Since the Hausdorff topol ogy Too induces on EP + I a topology weaker than Tp +1 ' it follows that M is also relatively compact in E.

Every (LN*) space E is reflexive (i. e. the strong bidual E " of E is topologically isomorphic to E). 5.4. 10. COROLLARY.

is a

In fact, E being the inductive limit of a family of normed spaces

barreled space, and this along with 5 .4.9 . , implies that E is re

flexive.

5.4. 1 1. COROLLARY. Eve ry (LN*) space is complete. PROOF. By the theorem there exists a se quence (Cn ) of bounded sets in E such that every bounded set H in E is contained in one of o

the Cn • This implies that the polar sets Cl ' C2 ,

•

tal sequence of neighborhoods of 0 in E ' which is

form a fundamen-

countable. Hence

is metrisable and as comple te . • E'

E is isomorphic to E " , it follows that

E is

Remark. The (LN*) spaces turn out to be "Schwartz spaces", according to the terminology of Grothendieck. They can be charac terized as the strong duals of the Schwartz metrisable spaces . Ob serve, however that for a direct definition of (LN*) spaces, as well as for their application to define directly the topology in spaces of distributions , the preceding theorems are needed, the results of Grothendieck being insufficient. For further information, see the references at the end of the chapter. 5.5. Topology of !!lJ (I), when I is a compact interval

In order to arrive at our goal, we still need a criterium concern ing a particular case of the concept of inductive limit introduced at the beginning of 5 .4. Let E be a normed space, F a vector space (over the same field IR or C ) , and qJ a linear mapping of E onto F. It is eas ily seen that the strongest topology on F making qJ continuous (the so called image top of the topology of E by qJ) can be defined by the semi-norm corresponding to the set qJ (B) where B is the unit ball in E. Then, F becomes a seminormed space, which is a normed space if! the kernel N( = qJ-I (O)) of qJ is closed in E (which is a necessary and sufficient condition for the set { O } to be closed in F) . This being so, we let I be a compact interval on fR. Then C(I) is a normed space according to the usual definition of norm recalled in n 5 . 1 . On the other hand, we have seen that the operator D for n = 1 , . . . defines a linear mapping of C(I) onto the space Cn (I) of distributions of rank s n. In these circumstances it is natural to consider the space Cn (I) provided with the image topology -z: of the topology T of C(I) n by means of D . Now, the kernel of Dn (i.e. the set of all functions qJ n such that D qJ = O in C) is the set ClPn ' which as we have seen is closed in C(/) (cf. remark to theorem 5 .2.2.). Consequently : 5. 5 . 1 . The

e to r space C/I) with the topology

v c

is a normed space.

Observe that now we have, both topologically and algebraically:

Besides, it is easily seen that:

5.5.2. A sequence (fk) of distributions on I converges to a distribu tion g on I in the 1'n topology if! exists a sequence offunctions Fk in n C(/) and afunction G E C(/) such that fk = Dn� for all k, g = D G and �� G uniformly on I. Now, we are going to prove that:

5.5.3. The normed spaces Cn (/) , n = 1 , 2, . . form a regular sequence according to definition 5 . 5 . 1 . PROOF. Remember that the unit ball B in Cn (/) , is the image n of the unit ball B in C(/); that is .

Now set

Then for all tP E B ' and all x, x + h El,

This shows that B ' is equicontinuous on the (compact) interval I ; B ' is also bounded, of course. Hence, according to Ascoli ' s theorem, B ' n+ i s relatively comp act in C(/) . But D 1 defines a continuous mapping n of C(/) onto Cn +1 (/) . Consequentely, the set Bn=DnB = D + 1 B' is rela tively compact in Cn+ 1 (/) , for all n, and this implies that the sequence (Cn (/) ) is regular • .

81 Remember now that : 00

!?fl(J) = Coo (J) =

U Cn(J) ;

n= l

thus it is natural to consider the vector space

!?fl(J) provided with the

Coo , which is the inductive limit of the topologies of the nonned spaces Cn (l) . topology

Then, according to

5 .5.4. !?fl(J) is a

5.5.3. :

(LN*) space.

In particular from

and

5 .4.7.

5 . 5 .2. :

5 .5.5. The concept of convergence for sequences in the locally con vex spaces !?fl(J) is the same as the introduced directly in 5.2. 1 . 5.6. Topology of �(n), where n is an open set Consider a vector space E, a family

For this case, we need the concept of mapping of E onto over the sa�e field

(lR

�.

C ) and let

projective limit.

(�)a EA

lfJa be,

for each

Suppose that, on each

'X"a '

�

locally convex topologies on E for which each locally convex topology

there is one

'X" *

a linear

there is defined a

Then it is easily seen that among all

lfJa

is continuous,

weaker than all the others: this is called the

tive limit of the topologies To define

a EA,

of vector spaces

'X"a in E,

projec

with respect to the mapping

directly it is sufficient to observe the following:

5.6. 1. A filter 3= converges to 0 in E( r *) if and only if the filter converges to 0 in �( 'X"a ) for each a EA . Now let Q be any open set i n the restriction operator

PI is a

fR.

For any compact interval

linear mapping of

!?fle D )

onto

lfJa '

lfJa(3=) lC D, !?fl(J) .

But !?fl(J) has been defined as a locally convex space, as a (LN*)-space.

Hence, it is natural to consider the vector space .§2J(.Q) provided with

the projective limit of the topologies of the spaces iiJ(I) with respect to the operators PI . Then according to 5 . 6 . 1 . :

5.6.2. The concept of convergence for sequences in locally convex spaces .§2J(.Q) is the same as introduced in 5 . 2 . 5 .

I t must be observed however that the locally convex space §if(.Q) is not complete. For example, the sequence of distributions f

� 8(')(x n

k) on IR is a Cauchy sequence on each compact interval

1, hence on fR, but it does not converge to a distribution on fR .

Obviously, we can define on the space §if (.Q) of all global dis tributions on .Q a topology as we did for !?2J(.Q). Then it is readily seen that §if(.Q) is a locally convex subspace of §if (.Q), which is dense in §if(.Q) . Moreover, remembering that every space §if(I) (I a compact interval) is complete being a (LN*)-space, is easily shown that:

5.6.3. !if (.Q) is complete. S o !!lJ(.Q) can also be obtained by the completion of §J( Q). Finally, it can be proved that the preceding topologies on §J(I ) and §J(.Q) coincide with the strong topologies introduced by L. S chwartz in these spaces, considered as the duals of certain spaces of Coo functions . Remark. The (LN*)-spaces turn out to b e a special category of Schwartz spaces ; they can be characterized as the strong duals of Schwartz metrisable spaces. This category of spaces has been point ed out by the author in 1 952, first for the study of spaces of analytic functions (see references below) . But its application to spaces of distributions required some developments introduced in 1 954. The

83 (LN* ) - spaces occur in a great number o f situations i n functional an aly sis, as far as distributions and analytic functions are concerned.

by Gro thendieck, on

Th e s pecific properties of these spaces are not implied in the paper

(F)

and

(DF)

spaces , where the S chwartz spaces

were introduced . S ome research on (LN* )- spaces has been made by

Yoshinaga and a generalization of this clas s of spaces has been presented by Kaikov.

REFERENCES [ 1 ] A. GROTHENDIECK. Sur les Espaces (F) et (DF) . Summa Brasi liensis Math . 1 953 .

[2]

G . KOTHE. Topologishe Lineare Riiume. S pringer 1 960.

[3] D . A. KAIKOV. Inductive and Projective Limits with Completely Continuous Mappings. D o kl . Atad . Nank . S S S R (N. S . ) 1 3 6 ( 1 95 1 ) 984-986. (Math . Reviews 1 9-754) . [4] J . S . e SILVA. SuiJondamenti della teoria deiJunzionali analitici. Portugaliae Math . 1 95 3 . [5] J . S . e SILVA. Su certe classi di spazi localmente convessi importanti per le applicazioni. Rendiconti di Mat. e delle sue applicazioni, Roma (5) . lA ( 1 95 5 ) , 355-4 1 0. [6] R. YOSHINAGA. On a Locally Convex Space introduced by 1. S. e Silva. Journ. Science Hiroshima Univ. (A), 2 1 , n.O 2.