Projective-inductive limits of Hilbertian sequence spaces

Projective-inductive limits of Hilbertian sequence spaces

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Eijndhoven, van, S. J. L. (1987). Projective-inductive limits of Hilbertian sequence spaces. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 87-WSK-01). Technische Universiteit Eindhoven.

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by

S.J.L. van Eijndhoven

Eindhoven University of Technology,

Department of Mathematics and Computing Science, PO Box 513.

5600 MB Eindhoven, The Netherlands.

-2-IDtrodudioIi

In

the

paper

{EGK] the author

et

at

have

introduced the locally

convex

vector

spaces

SI\)(>\} and

T

~)O.

'n1ese

spaces are detemlined by a directed set 4> of Borel functions on JR1/;. n e Nand ann-tuple A := (AhA2> .••• All.> of strongly commuting self-adjoint operators. TIle space S4>(A)

is

an

inductive limit of Hilbert spaces, which is not strict in most cases. The space T~) is a

projecUve

limit of Hilbett

spaces.

The spaces SiI'I(A} and T O(A)

are in

duality. If

we impose

the so-caDedsymmetry condition on the set . , J:hen the space SCt(A) equals T 'f'(A) and T Ct(A} equals

S'P(A)' where 'P denotes a well-specified direct.ed set of Borel functions on Il". So in this case

both $ . ) aDd T 4t(Il> can be described both as an inductive limit and as a projective limit of Hilbettspaces.

If the II-tuple A has only a discrete spectrum then the spaces SC(A) and T 4I(A) are equivalent to

sequence

spaces labeled by the eigenvalues of A. In [EGK] no attention has been devoted to

this spacial case although the study of sequence spaces has gained growing interest during the

past

years. In

tms

respect. I mention the monographs {KG] and

tR1

of Kamthan and Gupta and of Ruclde.

In this

paper't

two

classes

of

sequence spaces are introduced.

They contain

a

lot

of interesting

examples.

One

class consists

of sequence spaces which

are

inductive limits of Hilbeltian seqUence

spaces.

The. other class consists of sequence spaces which are projective limits of Hiltleitian

sequencespaoes.

I discuss the topological properties of these sequence spaces, their interrelations. duality~ continuous linear mappings, topological tensor

products

and kernel theorems.

Further.

I study relations between standan:l KtSthe sequence space theory and the theory developed in this paper.

O. Notations and preliminaries

In this paper DJ denotes a countable set. As a standard example one may take DJ == IN or DJ == JNd. By ro( DJ) we denote the space of all sequences labeled by DJ. i.e all functions from DJ into q;. The real part of the set ro(DJ) is partially ordered by the usual ordering s; defined by

0.1 a:S;; b :¢::> 'Vje D : aO) s; bG) .

Thus ro(DJ) becomes a directed set. By O)+(DJ) we denote the set of all positive elements of

O)(DJ). So a sequence a E ro(DJ) belongs to O)+(DJ} if aG):?!: 0 for all j E DJ. Let A be a subset of DJ. Then XA E O)+(DJ) is defined by

[

0 if j

~

A •

0.2 XA

G)

== 1 if j EA.

In particular we put OJ == X{j} and 1

=

X D' Further, for each a E 0)( DJ) by I. we denote the sup-port of a,

0.3 I. (j E DJ I aG) ~ OJ .

Also we employ the standard notations «:DJ), Ip(DJ), 0 < ps; 00 and co(DJ). For each a E O)(DJ) we have

0.4 a E <I>(DJ) :¢::> I. is a finite subset of DJ

a E Ip(DJ):¢::>

L

laG) IP < 00

je l )

a E loo(DJ):¢::> ~UD laG)1 < 00

Je l:J

O<p<oo

The pointwise product a . b of a, b E ro( D) is defined by

0.5

a . bG)

== aG) bO) • j E DJ .

Correspondingly, for each a E ro(DJ) we define a-I E O)(DJ) by

0.6

for j ~ I. •

for j E I..

-4-For each m e co( I ) we introduce the multiplication operator

Mm

from co( I ) into co( D) by 0.7 Mp.a=ll-a, aeco(l).

We observe that for p. e 1",,(1), the linear operator

M"

maps liD) into '2(1) continuously. In

particular, if and only if 11 e cO<D), Mp. : 1₂(D) -:; '₂,(D) is a compact operator and if and only

if 11 e 11(D), M,,: '2(D) -:; /2(D) is nuclear (= trace class). i

Let 1)1 and D2 denote two countable sets. For each al e co(D1) and a2 e ro(Di),

al 0 az e co(DIX Dz) is defined by

Thus we anive at the subset co(D1) 0. co(Dz} of ro(D1xDz),

Finally we jntroduce the set IN.,

0.10

The set IN. can be seen as the space of all sequences with only a

finite

number of nonzero entries with values in IN. For each m e IN. we set

00

Iml

=

1:

m(n).

n=1

ro+(JD)

which we call

root sets.

Each root set produces an inductive limit and a projective limit of' Hilbertian sequence spaces. The topological properties of these inductive and projective limits can be translated into properties of the corresponding root set.

1.1. Definition.

Let p(lD) denote a subset of ro+(lD) which is directed with respect to the ordering::;; of ro(JD).

The set p(JD) is called a root set it if satisfies the following conditions: l.l.i. ~eD

3

aep(D) :

aU)

> O.

1.1.ii. There exists a set I c /Nco with the property that

I.

I m I-V < 00 for certain v E IN,

mel

The set I labels a collection {~ I mEl} of subsets of lD with the following proper-ties

~f1 ~'=0, u ~ lD

mel

To indicate the dependence of I on v we write Iv instead of I.

Further, by R (lD) we denote the collection of all root sets in the cone ro+(JD).

Remarks.

- Let p(JD) E R(lD) with associated disjoint splitting {~I mE Iv} such that 1.Ui is

satisfied. From 1.l.i and 1.Ui it follows that for each m E Iv there exists b E p( ID) such that ,inf bG) > O. The assumed directedness of the set p(lD) yields that for every finite

sub-JeOm

set E of Iv there exists b E p(JD) such that min inf bG) > O. In particular, if the index set Iv

meE ieOm

can be taken finite itself, then the set p( JD) contains a sequence ao E I co( ID) which is

bounded away from zero, i.e. 3E>O'Iie D :

lIoG)

~ E, such that

We emphasize that the sequences in p{lD) are not assumed to be increasing or decreasing (cf. [KG], p. 283),

In this section we study the col1ection R (D) of root sets. First we introduce an ordering

6

-1.2. Definition.

Let PI(lD) and Pz(D) belong to R(D). Then PleD) is said to be subordinate to piOJ),

nota-tion Pl(OJ):::; P2(OJ) if the following condition is satisfied

(a:s; C b means I c

J!

and b-1a:S; C 1).

The root sets PleD) and P2(D) are said to be equivalent. Pl(OJ) - P2(OJ). if

Pl(D):::; P2(D)

and

pz(D):::; PleD) .

1.3. Lemma.

Let Pl(OJ) and PiD) belong to R(D) with Pl(OJ) ~ Pz(D). Then Pl(OJ):s; pz(D).

Proof. Trivial.

Next we introduce the so-called cross operation # in R{D}.

1.4. Definition.

o

Let p(OJ) E R(OJ). Then the subset plI(D) of ro+(D) consists of all U E ro+(OJ) with the

pro-perty that

'iaep(D) :

a·

U E J",,(D) .

1.5.

Lemma.

Let p(OJ) E R(OJ). Then p' (D) E R(D). So the I-operation maps R(D) into R(OJ).

Proof. The set p'(D) is directed, because for all U E p'(OJ) and all v E (l(D), the sequence

u+v belongs to p'(OJ) and U+v~ u. U+v~ v. Moreover. 3j E p#{OJ), JED. whence condi-tion loU is trivially fulfilled. We prove condicondi-tion 1.Ui.

Let {Om 1m E Iv} denote the disjoint splitting of D corresponding to p(D) as indicated in 1.Ui. Let U E p#(OJ). For each m E Iv there exists bm E p(OJ) and C > 0 such that

So the sequence U· XQ

m belongs to loo(OJ). Therefore we can define v E ro+(OJ) by

v

=

I

1m I (sup

uO»XQ", .

mely JeQ,.,

L

Iml (supa(J)xQ",~ C

L

(inf(b(J))XQ", me I" meI"

for some C > O. Thus we obtain

~D I v(j) a(j) I = sup I m I .sup u(j) .SUll a(J) ~

Je JJ me I" JE Q"m Je Q",

~ C sup sup U(J) inf bG) ~ C SUD U(j) b(j) < 00 •

mely jeQ", jeQm je D

Hence v e pit (D). By definition we get

Lim I (sup u(j» XQ",

=

L

(infv(j» XQ", . me I" leQ", me I"

Thus we obtain p#(D) e R(D). [J

1.6. Lemma.

Let p(D) e R(D). Then we have 1'(10):5 plt#(D).

Proof. Let fte 1'(10). Then for all ue 1'#(10), a-ue 100(10). Hence 3e 1'##(10). It follows

that p(D):5 1'##(10).

D

We arrive at the following important definition. 1.7. Definition.

A root set p(D) is said to be #-symmetric if 1'##(10) - p(D), i.e. if p#It(D):5 p(D).

1.8. Lemma.

Let Pl(D), p2(D) be root sets with Pl(D):$ piD). Then we have ~(D):$ pf(D). In

particu-lar, Pl(D) - pz(D) implies p[(D) == pI(D).

Proof. Let u e p~(D) and let a e PleD). Since Pl(D):5 P2(D) there exists be P2(D) and C > 0 such that a ~ Cb. It follows that n· a ~ Cn· b. Since U· be 1",,(10) we get n· a e I",,(D).

So n E pf(D), whence p~(D) t;;; pf(D).

0

1.9. Corollary.

Let p(D) e R (D). Then pit (D) is a #-symmetric root set.

Proof. By Lemma 1.6 we have p#(D) ~ 1'11##(10) and p(D):$ 1'##(10). So by Lemma 1.8 we

obtain p#illI (D) == pil (D).

\I

In the following lemma we present a subclass of R (D). It is proved that any element of R (ID)

8

-1.10. LemmjL

Let Iv denote a countable subset of N. with the property that

L

Iml-v < -. Let

{Om

t mE Iv}

-Iv

denote a disjoint splitting of D. Let y(1v) denote a directed subset of 00+(111') which satisfies

- "meIv3gey(ly): g(m) > O.

- "Ile),(!,,)

3

_ie),(!,,)

3c>o

"meIv : I m I g(m) S C g(m).

Put

p{D :y(1v»

= {

L

g(m)XQ".! g e ,,(1v)} . me Iv

Then p(D ;,,(111'» is a generating sequence set.

Proof. The proof is left to the reader. The verification of the conditions l.i.ii is

straightfor-ward. []

1.11. Lemma.

Assume the notations and conditions of Lemma 1.10. Put

Then p'(D ;,,(111'» - p(D

;1(111'»'

frQQf. Let w

e

1

(1v). Then for alI g

e

Y{lv)

( L

g(m)xQ",)' (

L

w(m)XQ".) =

L

g(m)w(m)XQ". E 1",,(D).

-4

-4

-4

So p(lD

;1

(Iv)} c plf(lD ;y{/v».

Conversely, let U E p'CD ;y(/v». We set

il

=

L

.SO]) (u(y) XQ". .

Melv JeQ.,.

Then il e pit (D ; y(1

v».

It follows that

m H ~t. (u(y) E

1

(I v)

1.12. Lemma.

Let Po(D) be a root set with associated disjoint splitting

[Om

I mEl v}, We define Yo([ v) by

Then "(0(1 v) is a directed subset of 00+(1 v). satisfying the conditions of 1.1.10 and

Po(JD) - p(ID ; "(0(1

v» .

Proof. Since poe ID) is directed, it follows that "(0(1 v) is directed. Further for each 01 E I v and each jo E

Om

there exists a E po(ID) with .su]> aG);=: aGo) > O.

JeQ",

Let a E Po(ID). Then there exists bE Po(lD) and C > 0 such that for all m E Iv (*) 1m I .suP aG):5:: C .inf b(j):5:: C .sup bG) .

Je Q", Je Q", Je Qm

So "(0(1 v) satisfies the condition of 1.1.10.

It is clear that po(ID) S p(ID ;yo(Iv». So conversely let a e Po(lD) then following (*) there exists b E Po(lD) and C > 0 such that

Hence p(ID ; "(I v» S po(ID).

The next three lemmas contain some technical results which we need in the formulation of . necessary and sufficient conditions that guarantee nuclearity of the inductive limits introduced

in the next section. 1.13. Lemma

Let p(ID) be a root set. The following statements are equivalent.

(i) For all a E p(ID) and all U E pil (ID), a· U E II(ID).

(ii) There exists a sequence Jl E II(ID) with Jl-I_{;=: 1 and p.-l.U E pil(ID) for all U E p#(lD).}

Proof. (The proof is inspired by the paper [MJ.)

(ii) => (i) Suppose there exists such a sequence p. E fl(lD). Let a E p(lD) and U E fl(lD).

(i) => (ii)

Then we have

because a· Jl-1

• U E 1",,( lD).

Let

{Om

I mEl v J denote the disjoint splitting of ID corresponding to p(lD). Since for each mEl v. XQ", E pil (lD) and XQ",:5:: a for certain a E p( ID) it follows that

XQ E II(ID), mE Iv. Hence d,.,

=

#

Om

= IIXQ

lit

< 00, m E Iv. We put

· 10·

Jl::;::

L

Iml-vd,;;lXQm' mE Iv Then

L

IJl())1

=

L

1m l-vd,;;1 h:Q.,nl < 00. je Jl) _me/v Hence Jl E 11(D).

Let U E

pit

(D). Then we have

Let a E p(D). There exists b E p(D) and C > 0 such that sup a())~ C inf b(j). me Iv'

jeQ"m jeQ",

Then we have

~ C

sUf. L

b(j)v(j)~ CBb'vOI .

mE v jeQ.,

It follows thaL a • Jl-1 • U E 1"",( lJ), whence Jl-1 • U E p' (lJ).

1.14. Definition.

Let p(D) be

a

root set. Then we define PP(D), p > 0, by

PP(D) = (aP : j ~ a(j)P I a e p(D)} .

1.15.

Lemma.

Let p(D) be a root set and letp > O. Then PP(D) is a root set, also, with

Proof. Let [Q,. I mel v) denote the disjoint splitting of D corresponding to p(D). Let a E p(D). Then there exists b E p(D) and C > 0 such that

1m I lip sup a(j) ~ C inf b()) , mEl v .

jeQ", jeQ",

I m I .su~ a(j)p

= (

I m I lip ,sup a(j»)p S; 0' ,inf b(j)P .

~~ ~~ ~~

(The function x ~ xP _{is monotoneously increasing.) Thus it has been shown thal}

PP(D) E R(D).

Also it is clear that for all v E ro+(D) we have

. Hence p'.P(D)

=

pP.II(lD).

o

In addition to Lemma 1.13 we have

1.16. Lemma

Let p(D) E R(D). The following statements are equivalent

(i) For all

a

E p(D) and all U E I''l (D),

a· u

E II(D).

(ii) There exists 11 E II(D), 11-1 _{~ 1 such that for all u E 1'# (D), 11-}1

• U E I''l (D).

(iii) There exists p > 0 such that for all a E p(D) and U E I''l (D), a· U E Ip(D).

(iv) There exists p > 0 and 11 E Ip(D), 11-1 ~ 1 such that for all U E I''l (D), 11-1 • U E I'll (D).

Proof.

(i) <=> (ii). This equivalence is stated as such in Lemma 1.13.

(ii) => (iii). Let 11 E II(D) with 11-1 _{~ 1 and 11' U E p#(D) for all U E p'(D). Then with} lC = Illlp, we have I ( E Ip(D), lC-1 ~ 1 and for all U E pll(D)

1(-1 • U S; 1l-(llp]-1 • U

whence 1(-1 • U e I'll (D), U E p# (D). It follows that

for all a e p(D) and U e p'l(D).

(iii) =:> (iv). For all ae p(D) and UE pll(D), a'UE lp(D) means for all b=aPE pr(DJ)

and v = uP E p#.P(D), b· v E II(D). So by Lemma 1.13 there exists l, E II(DJ)

with l,-I~1 such that l..uPEp'l,P(D) for all UEp'l(D). Hence

l,lIp'UE pll(D) for all UE p'l(D). Now observe that l,llpe Ip(D) and

l,-llp ~ 1.

(iv) => (ii). Let v E Ip(D) satisfy the stated conditions. Then we set 11

=

vf.!>]+1. It is clear that 11 E II(D) with 11-1 ~ 1. Moreover, by repeated application ofv we have

12

-Finally we introduce "tensor products" of root sets.

1.17. Definition.

Let D10 D2 denote two countable sets, let _{Pl(V 1) E} R(.I)I) and Pi.l)2) e R(D2). Then

Pl(D1) 0. Pi8)2) denotes the subset of 0)+(D1 x .1)2) defined by

1.18. Lemma.

The set Pl(D1) 0. P2(8)2) belongs to R(Dl x D2).

Proof. Let {~I m e Iv/} denote the disjoint splitting of .1), corresponding to the root set

p,(lD,), I

=

1,2. For each (mhmV E IVI x I,,? we put

L L

(Im\I+lm2If"t-Yzs (

L

Im1,-Vl)(

L

Imzl-v?).

~~¥~ ~~ ~~

Let a1 E Pl(lDt) and 82 E

P2(DV.

There

exists bi E P[(8)I) and b2 e pf(JDz) and C > 0 such

that

(1+ 1m/I) sup 8IU)S C inf biU). 1= 1,2.

jeQ,!. jeQ,!.

So the set Pl(8)I) 0. P2(lD2) satisfies Condition 1.Ui. It is evident that Pl(D1) 0. P2(D 2) is

directed and also that Condition l.U is satisfied. [J

Remark.

The

#-symmetry is not preserved under taking tensor products, in general. So if Pl(D1)

and

P2(D2) are I-symmetric root sets, then P1(8)I) 0. pilD2) need not be I-symmetric. Cf. Exam-ple 2.

2. Inductive limits of Hilbertian sequence spaces

To each root set P(lD) we link an inductive limit of Hilbertian sequence spaces. Naturally, on this inductive limit we impose an inductive limit topology. We introduce an explicit system of seminonns that generates the inductive limit topology.

Let a E w+(lD). Then by a·[z(lD) we denote the linear subspace of ro(lD) defined by

a ·/z(lD):::: Ma(lz(D»:::: {a·x I x E fz(D)} .

On a • lz( D) we introduce the inner product (. '-)a by

...

(a - x ,a' Y).

L

lA(j)x(j) yO) . j=!

It is non-degenerate, because a . x

=

a -x implies x(j) x(j) for all JED. The linear operator Ma maps the closed subspace lA -/z(lD) of Iz(D) isometrically onto the inner product space a -I z( D). It follows that a • I z( D) with inner product (. ,.). is a Hilbert space. We arrive at the following definition.

2.1. Definition.

Let p(D) be a root set. Then the space S p(D) is defined by

SpeD):::: u a-lz(lD). aep(D)

The Hilbert spaces a ·/z(D) establish an

inductive system:

For

all

a,b E p(D) with a:5 b we

have a-Iz(D) !:: b -[z(D) and the canonical injection i ab : a -[z(D) y b· Iz(D) is continuous. Hence on SpeD) we can impose the inductive limit topology brought about by the Hilbert

spaces a ·lz(D). So a set

n

c Sp(lD) is open if for every Xo E

n

there exists a convex balanced set U c SpeD) with the property that U () a .fz(D) is open in a ·1z(D). a E p(D), and xo+U

en.

Thus SpeD) becomes a locally convex topological vector space. In particular it fol-lows that all canonical injections ia: a '/2(D) y Sp(D) are continuous (cf. [Co], p. 119-121).

Let U E

p#

(D). Then for each a E p(D) we have u - a E I",,(D). So for all S E SpeD), s:::: a· x

with a E p(D) and U E pil(ID).

2.2. Definition.

On the vector space S p(D) we introduce the seminorms

pis) Ilu-sUz.

S E Sp(D)' U E pil(D).

14

-We prove the main result of this section.

2.3. Lemma.

Let p( D) be a root set.

(i) For all U E pll(D) the seminonn Pu is continuous on SpeD)' Moreover. ( l (pis)::: 0 Is E SpeD)} :::: (OJ. Hence Sp(lD) is a Hausdorff locally convex

lopologi-uep' (8)

cal vector space.

(U) Let 0 be

a

convex neighbourhood of the origin in Sp(l» with the property that for all

a

E p(D) the set 0

n a

"2(D) contains an open neighbourhood of the origin in a '[2(D). Then 0 contains a set V II

Vu := (s E Sp(l» I Pu(s) < I} .

Proof.

(i) Let a E p(D). The set (s E Sp(8) I Pu(s) < I} is convex and balanced. Moreover. for all

x e 1₂(D)

So the set {seSp(8)IPu(s)<l}n a'/2(D) is open in a·liD). It follows that

(s E Sp(JD) I Pu(S) < I} is open in SpeD)' whence the seminonn Pu is continuous. In addition, we have

n (s El SpeD) I Pu(s) = O} !:: ,n (s(j):::: 0 I S E S P(D)} :::: (OJ

~~~) ~8)

which implies that S P(D)

is a

Hausdorff topological space.

(ii) Let (Q.Jme1v denote the disjoint splitting of D associated to P(D) as indicated in Definition 1.1. As

we

have seen for each mEl v there exists a E p( D) such that AQm:S; Ca. Therefore for each mEl v there exists a E p(D) with

Xo.,.

'/2(D) !:: a '12(D),

For each mEl'll we put

rm=sup(p>OI[(SE XQ,..,12{D»1\ (lIslla<p)]:::;:'SE O},

Then fm > 0, mel'll'

Let a E p(D). Fix b E p(lD) and C > 0 such that for all mel'll

2 I m IV .SlW au):S; C ,inf b(j) .

JeQ". JeQm

Since {} ( l b -/2(D) contains an open neighbourhood of the origin in b -i₂(lD), there

It follows that ... . f b(j) 2elmlv (j) rm,e,m > C ,supa • JEQ,., JEQm me Iv. We define U E ro+(DJ} by Then (j) (j) ( 21mlv (j» C

suoa u = suo - - ,sup a <-,

JE bJ mel v rm JElQ,.,

e

Since a E p(DJ) has been taken arbitrarily we obtain U E p# (DJ). Next we show that for all 8 E S p(JD)

lin·

8

III

< 1 ~ 8 E n .

So let S E SpeD) with IIU'8112 < 1. Then 8 E

a

.12(DJ) for some

a

E p(DJ). For all mE Iv

we have

whence 21 m 1 v XQ,., • SEn. Further, there exists b E p( DJ) and C > 0 such that

1 m 1 v+l sup a(j) ~ C inf b(j) ,

jeQ,., jeQ,., me Iv. Let M E IN. Then we have

(*)

L

1IXQ,.,·slI'~~2v-2

L

IIXQm·slI;~~2v-2UslI;. mel". me/.,. Iml>M Iml>M We write 1 1 S

=

L - -

(21 m 1 v XQ • s) + (

L

21 m 1 v ) 8M mel.,. 21 m IV m mel.,. lmlSM Iml>M where SM

= (

L

_l-v )-1

L

XQ,.,· S • mely• 21m I mel.,. Iml>M Iml>M Since

16

-by (*) we have

Hence (SWM .. IN tends to zero in b .1:z(D). It yields

Mo

e IN such that sMo e Q. Since the set {m e Iv I Iml ~ Mo} is finite (cf. the condition on Iv). it follows that s can be represented

as

a sUbconvex combination of elements of Q. Convexity of .0 yields s e O.

0

2.4. Theorem.

Let p( JD} be a root set.

The locally convex topology on Sp(l». which is generated by the complete set of seminonns PUt

D E p# CJD). is equivalent to the inductive limit topology for S p(l».

In the following theorem we list some properties of the locally convex topological vector space S p(/D).

2.5. Theorem.

Let p(JD) be

a root

set.

(i) S p(D) is barreled.

(ii) S p(D) is bomological.

(iii) SpeD) is nuclear iff there exists p > 0 such that for all a e p(JD) and all 1l e p#(JD),

a'D E tp(JD).

Proof.

(i) Let V be a barrel in Sp(l»o i.e. a radial convex circled and closed subset of S fI(/D). Then

V n a '/a(D) is a barrel in the Hilbert gpace a ·i,.(D). So there exists an open neigh-bourhood of the origin .0. c V n a· I:z(D). because a ·1,.( D) is a Hilbert space. Follow-ing Lemma 2.3. V contains an open neighbourhood V U

=

(s e S p(/D) III u • slh < I} for a

suitable u e p#(JD).

(ii) Let

n

be a circled convex subset of SpeD) that absorbs every bounded subset of

B c

S

p(D). We have to show that Q contains an open neighbourhood of the origin in

Sp{/D)' For a E P(D) let B. denote the unit ball in a .lz(D). Then B. is bounded in Sp(/D)'

So there exists e > 0 such that e B a C

n

n a· 12( D). Now apply Lemma 2.3.

=::» Suppose S p(D) is a nuclear space. It means that for all u E pi (lD) there exist

v E

p*

(D) with v ~ u such that the injection j :

S

v y

S

u is nuclear. Here for any

W E

p*

(D),

S

w denotes the completion of

S

W'

with respect to the inner product (s,'S)'" = (w. S, W· 'S), s,s E Sw' It follows that for all

U E p# (D) there exist v E pI (D), v ~ 0, such that the linear operator

M_v-I•_u : 1

i.

D) ~ 12( JD) is nuclear. Hence V-I. 0 E II (JD).

Let 8E p(JD) and let DE p'eD). Take v~ 0, VE p#(JD), such that V-I'OE II(JD).

Then we get

<=) By Lemma 1.13 there exists .... E II(D) with .... -1 ~ 1 such that .... -1. U E p# (D) for all

U E p'l(D).

Let U E p# (D). Then v = .... -1 • U E p# (D) and v ~ o. The canonical injection j :

S

v y

S

u can be written as

j : S H

L

MIG) .... (j)(s, e j)V Ij

jeD

where (.,.)V denote the inner product in

Sv,

ej=v-1'Oj and Ij=U-1'Oj' Since .... e II(JD) it follows that the injection j is nuclear.

0

By Lemma 1.15, Lemma 1.16 and the previous lemma we have

2.6. Corollary.

Let p(JD) be a root set, p > O. Then SpeD} is nuclear iff SPP(lD) is nuclear.

2.7. Corollary. Let SpeD) be a nuclear space. Then the inductive limit topology for Sp(UJ) is generated by the seminorms p;, 0 E p'l(JD),

p;(s)=llu·sll..,=~ut> I uG)s(j}I , SE SpeD)' JE /l)

Proof. Let 0 E pi (D). It is clear that

II

0 •

sib

~

•

U·

slloo.

By assumption there exists .... E 12(D) such that .... -1 ~ 1 and .... -1. 0 E p'eD}. We get

It follows that the collections of seminorms {Po I U E p# (JD)} and {p; IDE p* (D)} arc

- 18 ~

In the family R(D)

we

have introduced a partial ordering :S. This ordering of R(D) imposes an ordering by inclusion on the eoIlection {Sp(J.) I P(lD) e R(D}}.

2.8. Theorem.

(i) Let Pl(D),Pz(D) e (D) with Pl(.D):s P2(D). Then Sp1(1)) ~ Sp2(1)) and the canonical injection

is continuous.

(ii) Let PI(D),Pz(D) e R(D) with PI(D) - Pz{D). Then SP1(ll) equals Spl(JI) as a locally

convex topological vector

space.

Proof.

(i) Let s e Spl(D)' Then s

=

al'x with 81 e PI(D) and x e lz(D). Since PI(D) ~ P2(D) there is 8z e piD) and C > 0 such that al s; Caz. We get

ail, 81 e

i",,(D). whence

By Lemma 1.8 we have ~(D):s pt(D). So for each U2 e p~(D) there exists

Ul

e

ptUD) such that U2 S; U1> whence

Therefore the injection j : S 1'1(1)) ~ S Pl(D) is continuous,

cf. Lemma 2.3.

(U) ObselVe that PlOD) - piD) means PleD) ~ pz(D) and P2(D) ~ PleD). So by (i).

S PleD)

=

S p,JD) and the identity i : SY1(D) --+ Sy,,(1» is a homeomorphism.

0

2.9.

Corallin),.

(i) We have S P(D) ~ Sp##(D) where the canonical injection Sp(D) ~ Spl#(/O) is continuous.

(ii) Suppose P(D) is a I-symmetric root set. Then SpeD) equals Sp#l(D) as

a

topological

vec-tor space.

Proof. Cf. Lemma 1.6 and Definition 1.7.

o

2.10. Corollary.

The space Sp(lD) is nuclear iff Sp"(D) is nuclear.

Proof. The space SpeD) is nuclear if and only if there exists" e II(D).

,,-I

~ 1, with 11-I,uE fleD) for all UE p'(D). Since piI(D)=piIiI#(lD) the result follows from Theorem

2.11. Corollary.

Let Po(JD) be a root set with associated disjoint splitting {Om I mEl y} of JD. As in Lemma

1.12 we define y(/y) in m+(/y) by

Then we have

(Recall that p(JD;y(/v»

=

(1: g(m)lQ,.,

I g E y(/y)}.) m

In particular, it follows that the topology on Sp<JD} is generated by the seminorms

with U E p(JD;.., (I v»~.

0

In general a space of

type

SpeD) is not complete. We discuss the problem of completeness of SpeD) in Section 4.

3. Projective

limits of

HHbertiao sequence spaces

To each root set p(

D)

we

link.

a projective limit of Hilbe.rtian sequence spaces

T

p(.4!>r

3.1. Definition.

Let p{D) be a root

set

The set Tp(l) consists

of all

sequences t e ff)(D) for which a· t E 12(D) for

all a e

P(D}.

The space

T p(D)

can

be

seen

as a

projective

limit

of

Hilbert spaces.

To show

this.

to

each

t € T_{pCD )}

we

link the mapping pmj, : P(D) - t liD) defined by

projt{a)

=

a· t • a e P(D) .

For each

t

e

T p(1))

we

have

Conversely.

if

a :function

P : p(D) - t 'l(D) satisfies

+.

then we define

t E m(/D)

as

follows:

Let jeD. Take a E p(/D)

such that

j e I

and put

t(j)

=

a(j)-I (P (a» (j) .

The definition of

t(j)

does not depend on

the

choice of a e

1'(1) because P

satisfies

+.

We

have

P ::::: pro,h.

Ai>

the locally convex topology for

T p(1)) we

take

the

proje<:tive

limit

topology,

3.2. DefinitiQn.

On

Tp{D)

we impose the locally convex

topology

generated by

the seminonns q.. a e p(D)

It is the coarsest topology for

which all

evaluation mappings t ~ proMa), t e T p(D)'

a

E p(O),

are

continuous.

3.3.

Lemma.

Let

t

e

S p'{D)'

Then

t

e

r

p(I»_

ProQf.

There exists u

IS I''l (llJ)

and x e 1

2( D)

such

that t ==

u • x.

So for all a e p( D)

we have

a· u· x e 1,.(D) because a· U E 1",,(D).

lJ

We characterize

the

bounded subsets of T p(D)' This characterization turns out to be useful in·

3.4. Theorem.

(i) A set BeT p(D) is bounded iff the set

fa·

t i t E B} is bounded in (z(D) for all a E p(D}.

eii) A set BeT P{D) is bounded iff there exists U E 1'# (D) and a bounded subset B 0 of 12(D)

such thatB

=

{u-x I x E Bo}.

Proof. Statement (i) is a simple consequence of the theory of locally convex topological vector spaces.

(ii) ¢:) Let Bo c 12(D) be a bounded set. So there exists C > 0 such that IIxUzs; C, x E B o.

Let U E p#(D). Then for all a E p(D)

~) Let B be a bounded subset of T p(lD)'

Let

{Om

I m E Iv} denote the disjoint splitting of D with respect to P(D). For each

mEl v there exists a E p(D) and C > 0 such that

XQ...

S; Ca. So Xo". -t E 12( D) for all t E T p(D)' Put

Since XQ

m S; Ca for suitable C > 0 and a E p(D),

Put

We show that U E p#(D). So let a E p(D). Then there is b E p(D) and C > 0

It yields

S; C sup (supU'Y~ -tlD(inf bG)}S;

22

-and hence u e pll(D). Further, for each t E B

L

Ilml-v12

r;;;IXo..,

.tli~ ~ lm!"-Y.

mE/y • mely

~>IIJ

So the set B 0 defined by

Bo= (u-l·t I

t

e B)

is bounded in 1₂(D), (We have u-

l •

t;:::

L

Iml-

VJ2 r;;;lXo..,.

t.)

_/y. r"...o

3.7. Corollary.

o

Let tE TpCD)' TIlen there exists ue pll(lJ} and XE Iz(D) such that t=u·s:. Hence Tp(lD)

equals S 1" (D)

as

a

set.

IJ

In addition to the characterization

of

the elements of T p(D)

as

given in the previous

corollary.

we have

3.8. Lemma

Let Sp(JD) be a nuclear space. Then t E ro(lJ) belongs to Tp(D) iff there exists u e pll(D) and x E 1",,( JD) such that t :::; U • x.

Proof. Since S p(D) is nuclear, there exists J1 e 1₁(D). Ji-1 ~ 1 such that Ji-1 • v e p* (D) for all

v E pll(JD).

=:» Let t e T p(D)' TIlen t = u· x with u e

pi

(D) and x e 1 z(D). Observe that x e i .. ( /D).

4:) Let te ro(JD), t=u·x with ue p'(D) and xe loo(D). Put v=Ji-1.u and Y=Ji'X' Then

v E pi (JD) and ye 1 t(lD) c 12(D). Hence t = v· yeT p(D)' []

The following lemma says that for each bounded subset B of T p(D) there exists u E p# (JD)

such that B c u ·lz(JD) and the topology of T p(D) restricted to B is equivalent to the topology of u· 1₂(D) restricted to B.

3.9. Lemma.

Let B be a bounded subset of T p(D) which contains the origin.. Then there exists u E pit (JD )

and a bounded subset B () of 1₂(D) such that B = {u· x I x E B 01 and, also, with the property: A

net (ta) in B converges to zero iff fa

=

u • Xa where Xa e lz(JD) and Y Xa II ~ O.

B={v·xlxEBtl.

PutK:= SU)) IIxliz and define UE p*(lO) by xeD! U=

L

ImlXQ",'v mely and Bo c I

_z

(lO) by Bo= {

L

Iml-1XQ""xlxE Bd. mely

It is clear that Bo is bounded and B := {u'x I x e Bo}.

~) Suppose the net (ta) in B converges to zero. Let ta:= U

':xu

with Xa E B o. Let e > O. Take M E IN fixed and so large that

~

<

2~

e. Then for all

a.

we have

(*)

II

L

XQ""xall;S

~z

L

ImlzUXQm'xau;<tez.

mE/y. mel.,

Iml>M

Because of the conditions imposed on p(D), cf. Definition 1.1, there exists a E p(D) and C > 0 such that

L

u-1, XQ", < Ca.

mel,. ImtSM

Furthennore, there exists 0.0 such that for

a.

~

0.0,

n

a • ta

Iz

<

2~

e. So for

a.

~

0.0

Combining (*) and (**) we see that there exists 0.0 such that for all ex ~ 0.0,

II :xu liz

< ~- e. <=) Trivial since a' U E loo( ID) for all a E p( 10).

We arrive at the following important result. 3.10. Theorem.

I]

Let B be a bounded subset of T p(D)' Then B is homeomorphic to a bounded subset B 0 of

12( 10). The homeomorphism cl> : B 0 -+ B is given by

cl>(x)

=

u • x •

-

24-Proof. From Lemma 3.9 it follows d1at ~ exists U E p'(D) with tile t'bU.owing pmpenies:

- B

=

{u' x I x e B o} •

- A nct(ta) in B -B converges to zem iff theIe exists a net (x.) in Be-lJQ such that

fa

=

U -XII and 1I~lh --+ O.

Finally observe that for each net (ta) in B with limit

1

e B. the net

<itt

-1) tends U) rem in

B -B. 1]

3.11. Corollary.

(i) A subset K. of Tp(J» is

.compact

iff there ·exisIs II e p'(f) aad a compact subset

ICe

of

12(D) such that K .= {u -X I X e

Ko}.

Hence tile set K c: T pCD) is compact iff it is

sequen-tially

compact.

(ii) A sequence (in)ne Ii in T p(ID)

is a

cauchy

sequence

iff

there

exists • E :p' {D} and

a

CIW-chy sequence (xJnelV in '2(D) such. that

tn

= a-Sa. n E IN. So T 1'(1)) is sequentially

com-plete.

3.12.

Theorem.

(i) The projective limit T P(D) is oomplete.

(ii) The projective limit Tp(j») is

a Montel

space

iff

for

all

ae

pCD) and

ue

p*(I).

a· u e (:o(D).

Proof. Let (Q" 1m e Iv) be the disjoint splitting of I) corresponding to the mot set P{D).

(i) Let (ta) be a Cauchy net in Tp(D)' Then for all Be p(D)~ (a·ta) is a Cauchy net in

12( D).

In particular for

all m e J v the

net

CxQ,. •

fa) is

a Cauchy net in

I

I-

If).

We put

xm

= ~

Xo... -

fa

(/I-D}-limit)

and define t e m(lD) by

t=

L

xm·

melv

Let a e p(D) and let Xa::: lim a· til (liD )-limit). Then for all m E Iv. a·

xm

XQm' x ••

a

whence a· t=

1:

a· Xm:::

1:

to...x. ""

X. E liD). So t E T P(D) and t is the limit of the mElv mel ...

Cauchy net (tc:J.

(ij) A locally convex topological vector space R is called a Montel space if every closed and bounded subset of R is

compact.

::;.) Suppose Tp(D) is a Montel space. Let a e p(JD) and let u e p'l(D). Let (Xn)neN be a bounded sequence in lz(JD). Then the set (u·x" In e IN) is bounded in Tp(JD). So its

closure is compact and hence sequentially compact by Corollary 3.11. It follows that the sequence (u· xJnel1 contains a converging subsequence in Tp(D) and hence

(a • u •

xJne

11 contains a converging subsequence in lz( D). Since the sequence

(x,,)ne 11 has been taken arbitrarily. the bounded linear operator M .. _{u :} lz( D) -+ lz( D)

is compact Thus we obtain a· u e co(O).

<=) Suppose a· u e co(JD) for all a e p(JD) and u e p'l(D). Since

it follows that

XQ".'

u e eo( D) for all u e pit (JD ). mel v and consequent! y

XQ".

e co(JD). mel v- So # ~ is finite. It yields. cf. 0.4.

11=

L

Iml-1

XQ.,.

E co(O)·

melv

Let B denote a closed and bounded subset of T p{D)- Then there exists u E I''l (D) and a bounded subset Bo of 12(0) such that

B

=

(u· x I x E Bo} .

Define v E I''l (D) by v

=

11-1 • U. Let (tn)ne IV be a sequence in B. Then we have

where x" E B 0 for each n e IN. Since 11 E eo(D). the sequence {JI.. x,,)n contains a converging subsequence (JI.. x"t)le IV with limit y E 1₂(0). So the subsequence

(in.)te 11 converges in T p(D) to t

=

v· y. Since B is closed we have t E B. Thus it fol-lows that B is sequentially compact and hence compact by Corollary 3.11.

0

3.13. Theorem.

(i) Let Pl(0).p2(JD) E R(D) and suppose PI(D):5 pz(JD). Then T_p1(JD):2 T_p2(D) and the canonical injection

is continuous.

(ii) Let PI(JD),PZ(JD) E R(D) and suppose PI(JD) - Pz(D). Then T_p1(D) equals Tp'PD) as a

-

26-Proof.

(i) By Lemma 1.8 we have pf(D)~ pt(D). So by Theorem 2.8 we have Sp{(D)&;,Spf(lD)"

We have the set equalities T p2(D)

=

Spf(D) and Tpl(D)

=

Spf{D) fonowing Corollary 3.8.

Thus we get T p1.D ) ~ T PleD)'

Let al e PleD). Then there are 8z e Pz{D) and C > 0 such that 81 S Caz. So for all

t e T p1. D )

From this continuity of j follows.

(ii) We have Pl(D):5 pz(D} and piD):5 PleD). Hence Tpl(D);: Tp1. D ) and the identity

mapping i : T pI(D) ~ T P1. D ) is a homeomorphism.

o

3.12. Corollary.

(i) Tp"(D);: Tp(D) and the identity mapping i : TptI(D) ~ Tp(D) is continuous.

(ii) If p(D) is a I-symmetric root set. then the identity mapping i : Tp##(lD) ~ Tp(D) is a

homeomorphism, i.e. Tp(D) equals Tp##(.I)

as a

locally convex vector space.

Proof. We only observe that p(D):5 pll'l(D) and pll(D)

=

pllllll(D) whence the set equalities

T p(D) ;: S p'(D) ;: S p'''(D) ;: T p#t(D) is valid.

Remark.

If the set p{D) is countable, then T p(D) is a Frechet space. So if there exists a countable root

set Pl(D) such that plI#(D) - Pl(D). Then by the closed graph theorem and the previous

corollary we obtain that T plIeD) ;: T p(D)

as

locally convex topological vector spaces. In

4. Symmetric root sets

We start with the main statement of this section.

4.1. Theorem.

Let p(OJ) be a #-symmetric root set.

(i) S p(D) T p' (1D) as locally convex vector spaces.

(ii) Tp(D)

=

Spl(D) as locally convex vector spaces.

So for each I-symmetric root set p( OJ) both S p(D) and T p(D) are both inductive and projective

limits of Hilbertian sequence spaces.

Proof. Following Corollary 2.9 and Corollary 3.13 we have Sp(D)= Sp'#(D) and Tp(D) = Tpl#(D)

as locally convex topological vector spaces. Moreover, by Corollary 3.8 T p(D) = S r,t(D) as sets.

Consider the identity

Since p(OJ) 5 pi/II (OJ) for all a E p(D) the seminorm

So i is continuous. Also, since p## (OJ) 5 p( OJ) for all b E plill (OJ) the seminorm t H H b . t II is continuous on Tp(D) and hence ;-1 : Tp(D) ~ SpIeD) is continuous. Thus we have proved that

T p(D)

=

S p'(D) as locally convex vector spaces.

Since pi/ (OJ) is a I-symmetric set, also, we obtain S p(D) = S p##(D) = T pIeD) as locally convex

vector spaces.

0

Consequently, we obtain the following important results.

4.2. Corollary.

Let p(OJ) be a I-symmetric root set.

(i) SpeD) and Tp(D) are complete.

(ii) SpeD) and Tp(D) are barreled.

(iii) S p(D) and T p(D) are bomological.

(iv) S p(D) and T p(D) are Montel iff for all a E p(D) and U E p# (OJ), a· U E co(OJ).

(v) SpeD) and Tp(D) are nuclear iff for all a E p(D) and U E p#(OJ), a· U E i1(OJ).

:emm.

See Theorem 2.5 and Theorem 3.12. Further, we mention the following auxiliary

28

-4.3. Lemma.

The following statements

are

equivalent: (i) 31'EIl(D)'P.-~1 Vllep'(I» : 11-1, u E p#(lD).

(ii)

3

_{p.ell(D).p.-l~l} 'V. 1 #II bep"(D): 11- ,b E P (lD).

Emof.

(i) => (ii) Let b E p#ll (lD). Then b ' U E 1 ... ( lD) for all U E 1'# (lD). In particular, b '11-1 , U E 1.,,(lD) for all U E 1'# (lD), whence b ' 11-1 E p#ll (lD). (ii) => (i) Follows from (i) => (ii), because p#(lD) = p#llfl(lD).

4.4. Lemma.

Let p(lD) be a #-symmetric root set Then the following statements

are

equivalent:

(i) Vaep(D)Vuep'(D) :

a·

U E co(lD). (ii) V _{aeplf(D) uep##'(D):} V

_a·

_{U E Co{D).} Proof.

o

(i) => (ii) Let a e p#It(lD) and U E p#lllt(lD)

=

p#(lD). Then there exist bE p(D) and C > 0 such that a ~ C b. Hence we get a .

use

b •

u.

Since b· U E co( lD) the result

fol-lows.

(ii) => (i) Evident. since p#ll(D);J p(D) and pfl(D) = pm(lD).

o

4.5. Corollaty.

Let p(lD) be a I-symmetric root set Then S E co(D) belongs to SpeD) iff u-s E Iz(lD) for all

uEp#(D).

0

4.6. Coronary.

Let p(lD) be a I-symmetric root set Then the following characterizations

are

valid:

(i) Let B be a subset of Sp(D)' Then B is bounded in S 1'(1» iff there exists a E p(JD) and a bounded subset Bo of Iz(lD) such that B = {a-x I x E Bo}.

(H) Let K be a subset of Sp(I». Then K is compact iff there exists a E p(lD) and a compact

subset

Ko

of I z( lD) such that K

=

{a' x I x E B o}. Hence K is compact iff K is sequen-tially compact.

(iii) Let (sJneJII be a sequence in Sp(I». Then (sJneN is a Cauchy sequence iff there exists a E p( lD) and a Cauchy sequence (xn)ne JII in iz( lD) such that Sa = a -Xn, n E IN.

Proof.

(i) Following Theorem 4.1, Sp(D) equals Tp#(D) as a topological vector space. So by

Theorem 3.4 the set B is bounded iff there exists a bounded subset Bo of 12(D) and

be p#ll(D) such that B

=

{b. x I x e Bol. Now for all b e p#ll(D) there are a E p(D) and C>O such that b:S:Ca. So B=(b.xlxeBol={a.ylyeE o} with

Eo=

(a-1·b·xlxE Bol clz(D) bounded.

For (ii)

+

(iii) we apply Theorem 4.1 and Corollary 3.11 and proceed as in (i).

o

Remark.

In general the inductive limit SpeD) is not strict. It follows from the preceding corollary that for

I-symmetric root sets p(D), the space SpeD) is very much like a strict inductive limit (cf. [Co],

p. 122-123). 4.7. Lemma.

Let p(D) be a I-symmetric root set and suppose S p(D) is a nuclear space. See Theorem 2.5. (i) Let s E m(D). Then s e SpeD) iff there exist a E p(D) and x E loo(D) such that

s

=

a ·x.

(ii) Let s e ro(D). Then s e Sp(.1D) iff U· s e loo(D) for all u E pil(D).

(iii) Let tE m(lD). Then tE Tp(D) iff there exist XE 1",,(D) and UE pil(D) such that t

=

U ·x.

(iv) Let t E ro(D). Then t E T p(D) iff a· t E loo(D) for all a E p(D).

Proof. Since S p(D) is a nuclear space, there exists 1.1. E Il(D), 1.1. ~ I, such that .... -1. U E pil (D) for all U E pil(D) and 1.1.-1 • a E p#ll(D) for all a E pilll(D).

(i) ::;:.) Trivial, because l

_z

(D) c I...,(D).

<:=) Let s

=

a· x with a e p(D) and x e loo(D). Then 1.1.-1• a E p#ll(D). So there exists b e p(D) and C > 0 such that 1.1.-1

• as; C b. It follows that b-l• a:S: C 1.1., whence b-1'a E II(D). Thus we get s = a·x

=

b· (b-I.a. x) and b-I • a· x E 1₂(D}.

(ii) ::;:.) Trivial.

<:=) Suppose for all U E pil(D), u·s E loo(D). So in particular, for all U E pil(D),

p.-l. U ' S E /..,(D) whence u·s E 1₂(D). From Corollary 4.2 it follows that S E S p(lD).

(iii) This is precisely Lemma 3.8. (iv) ::;:.) Trivial.

<:=) Let a· t E loo(D) for all a E p(D). Since pilll(D):5 p(D), we derive that a . t e 1",,(1) for all a e pllll (D). SO by (ii), t e S pIeD) == T p(l). because also S pIeD)

30

-Remark..

It should not be hard to prove that the results of the previous lemma remain valid if / ... (D} is replaced by I p( D} where any p > 0 may be taken.

4.8. Theorem.

Let p(D) be any root set such that S P(D) is a nuclear space. Then S p(D) = S P##(D) iff p(JD) is

I-symmetric.

In particular, if p(JD) is not I-symmetric, then there exists bE p##(D) such that bE Sp#l(Jl)} and b f£ S p(D)'

Proof. If p(D) is I-symmetric, then SpeD) = Sptl(D) by Corollary 2.13. Suppose p{D) is not I-symmetric. It means that there exists b E pi/it (D) such that for all a E p( D) and C > 0 there

exists j E IV with b{j) ~ C aG)o Put s = b • 1 = b. Then S E S p## (D) by Corollary 2.10 and Lemma 4.5. However, S~ SpeD)' Indeed, SE a·lz(D) for some aE p(lD) would imply b=a·]I; for certain]l; E lz(D). So h!;;;; i and x~ a-I. b. Hence we should have a-I. bE lz(lD), which yields a contradiction, because by assumption a-I. b ~ /",,(lD).

0

As we have seen, for a symmetric root set p(lD) the space Sp(l.) is complete and SpeD) is very

much like a strict inductive limit. The next theorem states that for nuclear SpeD) the #-symmetry condition is also necessary to obtain all these results.

4.9. Theorem.

The following statements are equivalent: (i) p( JD} is I-symmetric.

(ii) Each bounded subset of SP(lD) is a bounded subset of the ~ilbert space a .lz(JD) for

some a E p(D).

Suppose in addition that S p(D) is a nuclear space. Then we can add the following equivalent statements:

(iii) The space S p(l.) is complete.

(iv) The space S p(l.) is sequentially complete.

Proof. We prove the theorem along the following scheme (i) ¢:;> (ii) and, if S p(D) is nuclear, (i) => (iii) => (iv) => (i). (i) => (ii) Cf. Corollary 4.4.

(U) => (i) Suppose p(D) is not I-symmetric. Then there exists bE p##(lD) such that for all a E p(lD) and all C > 0 there exists j E IN with b{j) ~ Ca{j). It follows that if

h!;;;; i then a-I. b is unbounded.

Consider the set B = (b· x I x E ~(lD),

I

x

liz

S I}. Then B c <P(lD) c S p(D)'

So B is a bounded subset of SpeD)' Now suppose there were a E p(D) and K > 0 such that

Be {a-yIYE 12(D) •• yH2<K}.

It follows that

!!

~ B must hold and also

II

a-I. b·

xllz

~ K for all

x

E 4>(D) with U x H2 ~ 1. Hence a-I. bEL ..,(D) which yields a contradiction.

(i) ::;:. (iii) Cf. Corollary 4.2. (iii) ::;:. (iv) Trivial.

(iv)::;:. (i) Suppose p(D) is not I-symmetric. By Theorem 4.8 there exists bE p##(D)

such that b Ii S p{D)' Let (J .Jndl denote a countable number of finite subsets of

D such that J n c:;; J n+l and u J n

=

D. (The existence of the J n is guaranteed neN

by the countability of D.) We put

s,,=b-x.,. nEIN.

n

It is clear that

s"

E S p(D) for all n E IN.

Since SpeD) is supposed to be nuclear, there exists 11 E 1I(D). 11-1:?: 1, such that 11-1• U E p#(D) for all U E p#(D). So for each U E p#(D) and rn > n we have

h·

s.n -

U·

s"ni::::

L

I(u· b '11-1'I1Xi)12~ jeJm\Jn

~IICl·U.bN;'

L

1110>12.

jeD\Jn

It follows that the sequence (s,,)neN is a Cauchy sequence in SP(lD)' However. its

pointwise limit b does not belong to S p(D ). So the sequence (s,,)ne N does not

converge in SpeD) and SpeD) is not sequentially complete.

0

From the results of this section it follows that for #-symmetric root sets the spaces S p(LJ) and T p(LJ) are both inductive limits and projective limits of Hilbert spaces which are in strong

dual-ity. See also the next section. So the I-symmetry condition is very powerful. In this context we mention the following result

4.10. Theorem.

Let 1'( ID) be a countable root set. p( D) :::: {a... I rn E IN} . Suppose am S; lIm+l for all m E IV. Then p(DJ) is #-symmetric.

32

-Proof. We prove that the assmnption that p(l) is not I-symmetric leads to a contradiction. So suppose p(l) is not '-symmetric, i.e. there exists b e plli/(I) such that

So in particular

Thus we can find a sequence (j~ N with the property that

For each j E I) we set

I(j)

=

{m E N I jm

=

j} . Then we have

u I(j)

=

IN .

JED

Sten.l.

We show that I(j) is a finite set for each j e I). Let me l(j). Then jm = j and by (*), b(j) > m 8m(j). Since p( D) is a root set there exists lllo e IN such that llmo(j) > O. So if m ~ mo, we have

whence

We thus find

So the set I(j) is finite. ~.

Uh

=

(j E ID I IG) ~ 0} .

Then because of the finiteness of IG) and because of (***) the set 1D1 is infinite. For each

j E ID 10 let ~

=

min

(lG».

Since I(J) n

IGJ

= 0 for j ~ j' we get Ilj ~ III and therefore the set

{Ilj I jeD I} is an infinite subset of IN. Now we define v E 0)+( ID) by

Further we set u = V-I.

~.

We prove that u E pi(ID).

if j E 1D1 and 811jG) > 0 •

if j e 1D1 and 811jG)

=

0 •

ifje 1D1

To this end, let m e IN be fixed. We consider the following cases: - If j E ID \ ID, then u(j)

=

0 whence (a.n • u)G)

=

o.

- If j E ID h m S Ilj and alljG)

=

O. then a.nG) == 0, whence (am' u)G) ==

o.

- If j e 1D1' m S Ilj and ~jG) > 0, then we have (8m' u)G)

=

a.nG)(811j(j»-l S 1.

It remains to consider the case j E ID 1 and m >~. However, there is only a finite number of elements j E 1D1 for which J!j < m because Ilj ~ III for

Y.j

E _{1D 1.} Thus we get

~.

We prove that U· b

e

1<>o(ID).

Let j E 8)1. Then j

=

jll' and so b(j) > J!ja",(j) because of (*):

J "'J

- If alljG) == 0 we have

(b· u)(j) == b(j) Ilj buTl == Ilj .

- If all,G) > 0, we have J

So for all JED 1 we find {b· u)(j):2: Ilj. Since the set (Ilj I JED

d

is infinite the sequence b • u

is unbounded.

34

-Remark.

The above proof has been discovered by A. Kuylaars. In his master's thesis [Ku] the operation # is introduced for arbitrary subsets of 00+(1). We mention the following result:

Each countable root set p(l) is Isymmetric.

-To this end, we observe that each countable root set p( I ) is equivalent to a root set

p(

I ) )

which satisfies the condition stated in Theorem (4.10). Indeed, let p(lD) = (8m! mE IN}. Then m

the set

p(

lD)

=

{L

8k I m E IN) satisfies the requirements.

S. The pairing of S p(lD) and T p(JIJ." their duality Let p(JD) be a root set.

On the product space Sp(JIJ) x Tp(JIJ) we introduce a sesquilinear fonn <, ,'>' To this end, let

S E Sp(l» and t E T 1'(1». Then 8

=

a 'x for some a E p(D) and x E 12(D) and t

=

u' y for some u E pit (D) and y E h(JD). So

(*)

L

Is(j)t(j)I

=

L

la(j)u(j)x(j)y(j)1 ::;

lIa·ull ...

Ux·yfl1· je JIJ je D

Therefore we put

<S, t>::

L

s(j) t(j) , _SE Sp(I». tE Tp(D)'

jel)

Following (*) for all S E SpeD) and t E Tp(D)' <S,t> is well-defined. For all • E p(JD) with

S E •• 12( D) we have

<S,t> = (a-1• S,.· t:h, t E Tp(I».

Similarly for all u E pit(D) with t E u·12(D)

<S,t> = (u· 8,U-1, t:h, 8 E SpeD) •

We arrive at the following representation theorem. 5.1. Theorem.

(i) A linear functional I on S p(JIJ) is continuous with respect to the inductive limit topology for S p(D) iff there exists

to

E T p(JIJ) such that

1(8)

=

<s,to>.

(ii) A linear functional m on Tp(D) is continuous with respect to the projective limit topology

for T 1'(1» iff there exists So E S p(DJ) such that

met)

=

<So, 1> .

Proof.

(i) *) Let toE Tp(JIJ). Then

to=u·Xo

for some UE pit(JD) and XoE i₂(JD). Thus for S E S p(JIJ) we obtain the inequality

36

-Let I be a continuous linear functional on S p(lD)- It means that there are

U1> • _ •• Ull. e pi (V). and C > 0 such that

n

Il(s)1 ~ C

L

IUl'slz. s e SpeD) •

h'I

Since the set p# (V) is directed we can take U E p# (lD) with u:2: Ub k 1 •...• n. So by (*)

Il(s)1 ~ Cllu,slh. _{s e} SpeD) •

with

C

=

nCo The linear space Du:::: {u' sis E S p(JD)} is dense in X_y'12(lD). Define the linear functional I u on Du by

lu(u·s):::: l(s) • se SpeD)'

(We note that 0 ' S

=

0 implies I(s):::: 0). By (**) for all x E Du

So there exists XC) e Xl!' I

_z(

lD) such that

lu(x) (x,Xoh. x e Du. It follows that

/(s):::: lu(u, s) = (0' S.Xoh =

<s.to>

with to

=

U • XC) e T p(D)'

(ii) '*) Let So e SpeD)' Then So= a·Xo with Xo e Iz(lD) and a E p(V). So the continuity of the linear functional t H <So. t>. t e T p(D). follows from the inequality,

Let m be a continuous linear functional on Tp(D)' Directedness of p(V) yields

a e p(lD) and C > 0 such that

!m(t)1 ~ C!la·tUz.

Repeating the arguments of 5. 1. (i) '*) yields Xo e X~ .lz(V) such that

met) (a·t,Xo}z=<So,t>

with So = a . Xo e S p(D).

o

to introduce the weak topologies a(Sp(D).Tp(6J) and a(Tp(JD).Sp(D» for Sp(JD) and Tp(JD)'

respec-tively. The following theorem is a generalization of the Banach-Steinhaus theorem for Hilbert spaces.

5.2. Theorem (Banach-Steinhaus).

(i) Each weakly bounded subset of SpeD) is bounded.

(ii) Each weakly bounded subset of T p(D) is bounded.

Proof.

(i) Let B be a weakly bounded subset of Sp(1D)' Let ue pll(D). The set {u·slse B} is

weakly bounded in the Hilbert space 12(D). whence nann bounded in I,.(D). It follows

that B is a bounded subset of SpeD)'

(ii) Let V be a weakly bounded subset of T p(JD)' Let a e p(D). Then as in (i) the set

{a· tit e V} is weakly bounded and consequently nann bounded in lz,(D).

Conse-quently, V is a bounded subset of T p(JD)'

o

In the next theorems we characterize weak convergence.

5.3. Theorem.

Let (t,,)ne.IV denote a sequence in T p(JD).

(i) The sequence (t,,)neN converges weakly to zero iff for all a e p(JD) the sequence (a -t,,)ne IV converges weakly to zero in 1z,(JD).

(U) The sequence (t,,)ne N converges weakly to zero iff there exists U E pil (JD) and a sequence (X.JneN in 12(D). which converges wealdy to zero in 12(D), such that

t,,=u-Xn,ne N.

Proof.

(i) We have

t" ~ 0 weakly in Tp(D) ~ "isesP(D): <s.t,,> ~ 0

~ "iaep(D) "ixe1i.D ) : (x, a· tJ ~ 0 .

(ii) <=) If in=u·xn , ne lN, with ue pil(JD) and (XJneN a weak null sequence in iz(JD),

then (a· t,,)"eN tends wealdyto zero in 12(JD), because a· U e lcoCJD).

=» The sequence (tJneN is weakly bounded. So by Theorem 5.2 it is bounded and there exists it e pil (JD) and a sequence (yJndl in Xii -/2(JD) with K := SUP II Yn

liz

< QO

nel'>l

38

-U

=

L

Iml

XQ",'u

and Xn

=

L

Iml-1XQm' Yn'

mel.. _{mel v}

Thent,,=u.xnandxne lz(D),ne N.

Let y e lz(D). e > 0 and Il10 e If with mo> 2K U yllz. Then for all n e IN we have

e

(*)

L L

'xnG)l z

=

L

Iml-z

L

IYnG)12~

Iml>mo jeQm Iml>mo jeQ",

We put

C1

=

u

Om.

Then u-1_{• "I~ • Y E SpeD)' because of the conditions stated in}

Iml~mo '''\Ie

Definition 1.1. So there exists

no

E IN such that for all n >

no

Since u-1

• U • Xn = xn. it follows that

Summarizing we get for n ~ no by (*) and (**)

1

e

<1.

e+IYI.- -

=e

2 2

Iyl

.

o

Similar to the characterization of weakly convergent sequences in lz(D), we have the follow-ing characterization.

5.4. Corollary.

Let (t,,)ne IV denote a sequence in T p(D)' Then the sequence (t,,)ne IN converges weakly to zero iff

(t,,)ne IV is a bounded sequence and for all j E ID, (t"U»ne IV is a null sequence in q; .

IJ

5.5. Corollary.

T p(lD) is a Montel space iff each weakly convergent null sequence in T p(D) is convergent (in the sense of the projective limit topology for T p(D»'

Proof.

¢:) Let a E p(D) and u E p#(D). Let (XnAteN denote a weakly convergent null sequence in

12(D). Then (U'xJneN is a weak null sequence in Tp(l/). So (U·xJneN converges to zero in the projective limit topology of T p(l/). It follows that (a· u • xJne II tends to zero in 12( D ) in nonn sense. Because (xJneN has been taken arbitrarily. we obtain a' U E Co(D).

By

Theorem 3.12. T P(D) is a Montel space.

=» Let t" ~ 0 be weak in T p(JD)' Then there is U E p# (D) and a weak null sequence (xJne II in

lz{ D) such that

tn

=

U • Xn• Since T p(D) is a Montel space, for all a E p( D) and all U E pi/CD). a· U E Co(D). i.e. M •. u is a compact operator from 12(D) into 12{D) for all

a E p{ D). U E pi/ (DJ). It follows that

B

a •

tn

Hz

=

II

a . U • Xn 112 ~ 0 for all a E p( DJ).

0

Next we present the corresponding statements about weak sequential convergence in S p(l/). The proofs are left to the reader.

5.6. Theorem.

Let (s.,)nell denote a sequence in Sp(JD)'

(i) The sequence (s.,)nell converges weakly to zero iff for all U E pi/(DJ) the sequence

(U'Sn)neN converges weakly to zero in 12(DJ).

In addition assume that p(D) is a I-symmetric root set.

(ii) The sequence (Sn)neN converges weakly to zero iff there exists a E p(DJ) and a weak null sequence (Xn)ne N in 1₂₍ D) such that s., = a . Xn. n E IN.

0

5.7. Corollary.

A sequence (s.,)neN in SpeD) is a weak null sequence iff the sequence (Sn)neN is bounded and for all j E DJ the sequence (s.,G»ne N tends to zero in q: •

5.8. Corollary.

Suppose p(DJ) is a I-symmetric root set. Then SpeD) is a Montel space iff each weakly

conver-gent null sequence in SpeD) is convergent (in the sense of the inductive limit topology of

S p(l/)).

We finish this section with a discussion of the strong topology for S p(lV) and T p(l/)'

respec-tively.

The strong topology ~(S p(D). T P(l/)) is the locally convex topology for S p(D) determined by the

seminonns

Pv(s)

=

SUD ks, t> I

te9

where V runs through the family of bounded subsets of Tp(D)' Similarly, the strong topology