Results in MathematicsVol. 21 (1992)

0378-6218/92/040289- 10$1 .50+0.20/ 0(c) 1992 Birkhiiuser ver Iag, Basel

BARRELLEDNESS A ND BORNOLOGICAL C ONDIT IONS

ON SPACE S OF VECTOR-VALUED p- SI M P LE FUNCTIONS

SANTIAGO D IAZ, L ECH DREWNOWSKI , AN TONIO F ERNANDEZ,

M IGUEL F LORENCIO AND P EDRO J . PAUL

ABSTRA CT. Let 5(1' , E ) be the sp&<:e of (classes of I' -"" e. eq u...l) s imple functi ons definedon a (non -trivial) meas ure spac e with v..lues in ... loc..lly convex space E. The following!'<'Suits hold : 5(1' , E ) is quasi-barrelled (r ,,",p. homological) if end onl y if E is quasi-b...rrelled( resp . hom ologic ..l) ..nd E'({3(E' , E )) has the property (8) of Pietsch ; 5( 1', E ) is barrelled if..nd only if 5(1' ,K) is ba rrelled ...nd E is barrelled and nuclear; 5 (1', E) is never ult rabo molo­gical ; ...nd 5(1', E ) is .. DF - sp&<:e if and onl y if E is ... DF-sp&<:e.

Several authors have studied barrelledness-type condit ions on the space of simple func­tions defined over an algebra of sets with values in a locally convex space, when this spaceis endowed with the uniform convergence topology. For instance, J . Mendoza [11] charac­terized when this space is quasi-barrelled or bomological , and a different approach to thisproblem was presented by A. Defant and W. Govacrt s {2, Pro ps. on p. 354]. Moreover,F. J . Freniche [6] gave necessary and sufficient condit ions for this space to be barrelled. Insome contexts, e.g. in integration theory , it is natural to consider simple funct ions definedon measure spaces. Thi s means that one should consider not individual functio ns, bu tclasses of functions that eoincide almost everywhere (with respect to the measure in ques­tion). The purpose of this paper is to study barrell edness-type condit ions in this moregeneral setting. We address this problem by means of two different approaches: Stonerepresentat ion theorem and injective tensor products. Although using tensor prod uctsis a longer way, it provides some additional results. We have decided to include bothapproaches because we think that each of them is interesting in its own way.

We star t by establishing our framework and recalling main definitions. We refer thereader to H. J archow's monograph [8J for the tenninology ab out local ly convex spaces. Awide and deep study of barrelledness and relat ed concepts ca n be seen in P. Perez Carrerasand J . Bonet 's book [12].

Framework. Let n be a nonempty set an d E a Boolean ring of subsets of n, i.e. the emptyset is in E, and the difference, finite union and finite intersection of elements of E belong

1991 Mathem atic6 Subjecl Cla66ifical;on. Prim...ry 46A08, 46E30. Secondary 28805 .Key word.. and phrw e, . Spaces of simple measur ...ble funct ions . B.. rrelled spaces. Bor nologic...1 sp &<:es ,

Vectcr-v ..lued set functions.This research has been suppo rted by L.. Con...ej.-ria de Educaci6n y Ciend .. de I.. J unt a de And..luci... .

290 Diaz et al .

again to E (but n itself need not be ill E). Let E be a (Hausdorff) locally convex spaceover the field K of real or complex numbers, and denote by Q the family of all continuousseminorms defining the topology of E, and by 111) the family of all absol utely convex andclosed zero-neighbourhoods of E .

A E-simple function 'f' : n -+ E is one with a finite number of non-zero values, each ofthem taken on a set of E. Following [6] and 111] we denot e by S(E,E) the vector spaceof al l E-simple functions defined on n with values in E. It is clear that every non- zerofunction 'f' E S(E, E) can be written in the following form

( . ) ~(-) ~ LXA; (-)x; ,; :1

where the X i E E (i = 1,2, . .. , n) are non -zero and different from each other, and theAi E E (i = 1,2, ... , n) are nonempty and pairwise disjoint (ali usual, XA st ands for th echaracteristic funct ion of a set A). Thi s representation is unique (up to permutations ofthe index set {I , 2, ... , n } ). Note that (A;):': I does not necessaril y cover fl .

The space Sp::::, E ) is endowed with the uniform convergence topology. A fundamentalsystem of seminorms for this topology is given by the seminorma vc defined by

'E (t XA,Xi) := sup{q(x;) ; i = 1,2 , ... , n},.:1

when q runs through the set Q.

Now, an d in what follows, let IJ : E -+ [0, +001be a finitely additive, non- zero measuredefined on the ring E (we allow sets of infinite measure). We identify two funct ions ofS(E, E) if they are equal almost everywhe re with respect to J-I (shortly, It-a.e.), i.e. if theycoincide except on a set of E with J-I- measure zero. Th is is an equivalence rela tion and wecan consider the quotient space associated to it . We first prove that we have a Hausdorffquotient (compare with [4, Lemma 2 on p . 112]).

Proposi t ion 1. Let NJ' be the subspace of all func tions in S (E, E) that are J-I~a . e. equalto the zero function. Then N" is closed .

Prooj. Let l{) be a simple function such that l{) rf:- NJ'. Then, we can write l{) in theform (*), requiring in addition that J-I (At} > O. Since X I f- 0, there is a continuousseminorm q E Q such that q(x J) > O. Consider the following open neighbourhood of '.P,V := {tb E S(E,E); qr.(If - cp ) < q(x t}/ 2}. It is enoug h to prove that V nNJ' is empty.Take T/J E V. According to (*), T/J can be written as tb = Lj':1 X Bj Y j with Yj i- 0 forj = 1,2 , ... , m . Let B = Uj: l Bj be the support of If; then IJ(AI " B) = O. To see thi s,suppose, on the contrary, that J-I (A I \ B ) > O. Since 1jJ- <.p takes the value - X I in Al \ B, wehave q(xJ )/ 2 > qr;( T/J -l{) 2=. q(- xd and thi s is a contradiction . Now, since IJ(A I "il ) = 0,we have that Jl(A I n B ) = Jl(A1 ) > O. Therefore, there must be an index jo such thatIJ(A1 n B jG) > 0, hence IJ(Bj . ) > O. Final ly, since Yj G i- 0, we conclude that 1jJ f/. NIP 0

Diaz et at • 291

Definit ion. The space 5( J.l, E) of «-ei mple (classes of) funct ions with values in E isdefined as the Hausdorff quotient

endowed with the corresponding quotient topology.

Let Eo be the ideal of Wnull sets in E. What we have defined is the quotient space5(E,E)j5(Eo,E), where the measure J.l does not play any essential role; we could havetaken any ideal Eo C E. Note, however, th at if Eo is any ideal in E, then Eo is preciselythe ideal of null sets for the measure J.l defined on E by setting J.l( A) = 0 if A E Eo, and= +00 otherwise, so that both situat ions are essentially the same.

We will not distinguish by notation between a simple fun ction and its J.l-equivalenceclass. In particular, using (..) above, for a lion- zero <p E S(p ,E ) we shall always choosea representat ive of the form <p( .) = :L::'l XA; ( ') Xi where the Xi E E (i = 1, 2, . . . ,n) arenon- zero and different from each other , and the Ai E E (i = 1, 2, .. . , n) are nonempty,pairwise disjoint, and Il (A;) > 0 for every i = 1,2, . . . ,no Bearing this rep resentation inmind , the quotient topology of S(Il ,E) can be given by the family of seminonns qoo definedas follows:

qoo(~) ' ~ inf(qE( ~ ) , ~ ~ ~ [u -u.e.j ]

= sup{q(x;) : i= I, 2, . . . ,n} (q E O).

The aim of this paper is to stu dy when S(J.l, E ) endowed with this topology is quasi­barrelled , barrelled, bomological, ultrabornological or a DF- space. There are some tr ivialcases, e.g. when E is finite or, more generally, when there is only a finite number of pairwisedisjoint set s with positive measure in E. In these cases, 5 (/1, E) is isomorphic to En forsome n E N and S(Il , E) is quasi-barrelled , etc ., if and only if E has the same property.To avoid these triv ial cases, we assume the following condit ion:

(C ) There is an infinite sequence (6.n ) of pairwise disjoint sets in E such that 1l(6.n ) > 0for every n = 1,2, . . . .

Our analysis of S(J.l ,E ) depends on a tensor product rep resentation where the factorsare the scalar-valued space S(Il ,K) and E. We refer the reader to (8, Ch. 16J, (9, §44]or [12, 11.4 and 11.5] for a det ailed st udy of topolo gical tensor products; however, let usrecall some definitions.

The e-topology, or inject ive tensor topology, and the s --topology, or project ive tensortopology, defined on F) 0 F2 are well-known. We are also interested in another topologyon the tensor prod uct ; namely, the A- topology considered by Hollstein [7] (t his topo logyis called doo in [3] and r- topology in [12, 11.4.371) and defined by means of a fundament alsystem of serninorms as follows. Let 0 ) and O2 be systems of seminorms generating thetopology of F1 and F2 , respect ively. For ql E 0 1 and sn E O2 , consider the seminonndefined on F1 119 Fz by

(qdhq' ) (' ) ' ~inf { sup q, (t "n,q,(yol)1..;1 9 ;=1

292 Diaz et a1 .

These seminorms (91 @),fh } (onn a fWldamentai system or seminorms for the ~-topology.

The corresponding locally convex space is denoted by F. C!h . F] . Since

for the corresponding !- and w-seminorms, the >.-topology is finer than the e- topclogyand coarser than the ",-topology.

A locally convex space F1 is said to be an e-space (see [71 . (12, Cb. 11J) if the e-topologyand the >.-topology coincide on F I 0 F] for every normed space Ft . The e-spaces playacent ral role in the commutability of inductive limits and c-tensor products. These spaceshave been also considered by A. Defant and W. Govaert s [31 in their study of barrelledneseand homological conditions on spaces of vector-valued cont inuous functions .

Theorem 1. The spaces S (J,l . E ), S (Il, K) 0~ E and S(p , K) ~h E are isomocpbic to eachother. In p lU"ticular, S(Jl, K) is an c-epece.

Proof . Consider the bilinear mapping 1,

•:= L XA.; CI';X E S(p ,E ),

i _ I

and let 1" : S(p, K) 0 E _ S(p, E ) be the associated linear mapping.

It is dear that each non- zero element z E S(p, K) 0 E has a representa t ion of the formz = :[:".1 XA; 0 Xi , where the Xi E E are nonzero an d different from each ot her, and theA; E E are pairwise disjoint with #J(A;) > 0 for i = 1,2" , . •n . From thi s it follows easilythat t - is injective. Clearly, t - is also surject ive and , therefore , S(p, E ) and S(p, K) €I Ecan be ident ified algebraically. For the topological ident ification of these three spaces, weonly have to show that the uniform convergence topology induced on S(p, K )0E is coarse rthan the a- topology and finer than the .\-topology.

Take U E Uo and z = :[:. 1XA; 0 Xi as before , If q denotes the gauge of U, then thereis some io E {1,2•. . . •n} such that

,~(I" (,» = ,~ (t XA'X;) ~ ,up{,(x,) ,i =1.2 .... . n} ~ ,(x.. ).•= 1

Now, denote by H t he linear hull of {XA ; : i = 1, 2, . . . , n} in S(p., K). The map 1 definedon H by 1(:[:=1 n;XA ;) := 0 ; 0 has norm 11 111 = 1 in H' . By the Hahn- Banach theorem,1 has an extension, that we st ill call 1, such that 11111 = 1 in 5 (1/, K), . Then, we can write

q~ (/' (x)) ~ , (x ,, ) ~ ,up{l(x...u)l , u E U' ) ~ ,up{1t.(XA..f){x,.u>1' u E U' }

s ,uP{It,(XA..9}(X;,u>1 ' 9E S(p.K)'. 11911 $ 1. uE U' } '

Diaz et, at • 293

Since the closed unit ball of S (p., K)' and U O are equicontinuous set s in S(p. , K)' and E' ,respectively, the latter expression is a continuous scminorm for the c-topology. Thi s showsthat the uniform convergence topology is coarser than the a- topology.

On the other hand, let q be a seminorm on E and II · ll be the nonn on S (p., K). Givenz = E~, X,,; 0 x ; as before, we have

(II· 110, q)(, ) $ 'up{11t. XA;O,q(,,)11' 10 01 $ 1, i ~ I' 2, ... ,n}= sup {Io;q(x; )l ; 10;1:::; 1, i= I , 2, ... ,n}

~,up{q(,;), i ~ I ' 2 , ... , n)~ qoo (t xA; ' ;) ~ qoo(r(,)).,=1

And this shows th at the '\~topology is coarser than the unifor m convergence top ology. 0

In order to develop our approach, we still need to recall the not ion of S '""- space givenin (3, DeL 1] or [8, p. 430]. A normcd space F is said to be an S '"" - space (with respect toF ' ) if there is a E R.with a<a < 1 such that for each n E .N thereare x l . x2, . . . •xn E Fand xLx~ , .. . , x~ E F' with

(1) (x;, x j) = 06;], II xdl :S 1 and IIxjll :s 1

"for all i, j = 1,2, . . . , n .

(2) I:1(,,,,' )1$ 1; = 1

for all x ' E F' with IIx' ll :::; 1.

Obviously, if a nonned space F contains an isometric copy of i': for every n E N. then itis an S '""-space. In particular . we have the following.

Prop osi t ion 2. S (p. , K ) is an S'"" ~space.

Proof. Let (6.n) be the sequence of pairwise disjoint set s given by condition (C ). Clearly,the subspace span ned by XAI ' X6 " .. . , X6~ is isometric to i':. 0

Corollary. Let E be the indu ctive limit of a sequence (E n) of locally convex spaces. ThenS ( p., E ) is tile indu cti ve limit of the sequence (S (p. ,En) ).

Proof. It follows from R. Hollstein 's result [7, Prop. 4.3], (12, 11.4.43]. 0

Some of the forthcoming result s arc given in terms of the well-known property (8) ofPiet sch [13, 1.5.5] (see also [12, 4.8.2] where it is called fundemental-r'<boundcdness).Thi s property was introduced in the berrelledness context by A. Marquina and J . M. SanaSerna (to) and it has been exploited by J . Mendoza (see [11] and references therein) andJ. Schmets (14] in the framework of spaces of vector-valued cont inuous functions (see also[12, 11.10.1- 31).

294 Diaz et al.

Theorem 2. S(~,E) is quasi-barrelled (resp. homological) if and only if E is quasi­barrelled (resp. bom ological) and E '{fi(E' , E » has the property (D) of Pietsch.

Proof tlia tenJor products , Use T heorem 1 an d Proposition 2 above , and the st a tements(b )",(e) and (e)",(d) of Theorem 9 in [3].

Proof via Stone repre"entation theorem. The idea here is to reduce to the situat ion studiedby J . Mendoza, namely to the space of simple funct ions defined on a Boolean algebra withno measure involved .

Let Eo be the ideal of JJ- null sets in E and consider the quotient ring 'B/Eo . By theStone representation theorem [5, Thm. 1 in §18.1], there are a nonempty set 0 ° and aBoolean ring E" of subsets of n" such that E" is isomorphic to B/Eo. Obviously 5(I-I ,E)is isomorphic to S(E* ,E ). Condi tion (e) ensures that P is infinit e.

Now let A- be the Boolean algebra of subsets of no given by

A · -= {A· C n·: either A · E E· or 0 · ", A- E E-}.

It is easy to see that every simple func t ion 'P. E S(A ·, E ) can be written in a unique (upto perm utations of the index set) way as:

"",. = L XA;X j + XB·X

;=1

where Ai ,Ai•.. . ,A~ E 1:", n· <, B" E 1:, and the values x, X l , x 2• ... ,x" E E are differentfrom each other . Define

P(lf'- ) = L: XA; Xi.; = 1

then P is a cont inuous linear projection from S(A ·, E) onto 5 (1:" E ). If A - = E., thenP = 0 and 5(A· ,E) 9::: S(p, E) . If A · :f:. E., then we have the following topologicalisomorphisms

Now, in either case , the theorem follows from J. Mendoza 's result s [11, Thms . 1.14 and3.3J for spaces of simple functions defined on a Boolean algebra. (In the second case , recallthat obviously, E is isomorphic to a complemented subspace of S(p,E).) 0

R emarks. A. Defant and W . Govaerts [2, p. 354] considered the following situation: LetS(X) be an infinite dimensional ring of bounded scalar- valued functions defined on a set Xthat is closed under complex conjugation , and let F be a lo cally convex space. Assumingthat S(X ) contains all constant functions, they proved that the space S(X ) 0~ F is quasi­barrelled (resp. bornological) if and only if F is quasi-barrelled (resp. bornological] andF' (fJ (F' ,F)) has the property (B) of Pietsch. Proceeding similarly to the fina l part ofthe proof abov e, one can prove that thi s result holds without the hypo thesis that S(X)contains all constant funct ions: work with the ring S(X) I!:l K generated by S(X ) and theconstant funct ions on X .

Diaz et al. 295

Theorem 3. S(p , E ) is barrelled if and only if S(Jl , K) is barrelled and E is barrelled andnuclear.

Proof via tensor product" . (<=) If E is nuclear, then the inject ive and projective tensorproducts of S (p , K) and E coincide [8, 21.2.1\ . The conclusion follows from Theorem 1and th e fact that the project ive tensor pro duct of a normcd bar relled space and a barrelledspace is barrelled [12, 11.2.2}.

(=» Suppose that S(p , E) = S (Jl, K) 0~ E is barrelled. Obviously, S (p , K) and E arebarrelled. Now, bearing in mind condition (e) , the linear subspace of S(Il , lK) spannedby {X6.M : n E N} is isometrically isomorphic to ¢i, the space of all sequences with onlyfinit ely many non-zero coordinates, i.e. S(p , lK) has a copy of ¢. Since S(p ,K) 0~ E isbarrelled , we obtain that E is nuclear by [G , 1.2].

Proof via Stone repre.Hntation theorem. P roceed as in th e proof of Theorem 2 to reduc eto the case of the space of simpl e fun ctions defined OIl a Boolean algebra with no measureinvolved . Then usc F . J . Freniche's resul t for these spaces [8, Cor . 1.5]. 0

Corollary . If E is a a -algebra of subsets of fl , tlWll S (p , E ) is barrelled if and only if Eis barrelled and nuclear.

Proof. If E is a a -algebra, then S(E, K) is always barrelled [12, 4.7.2]. Since S (/-I ,K) is aseparated quot ient of S (E ,K), then S (/-I , K ) is barrelled. Now, apply T heorem 3. 0

T heorem 4 . S(Il ,E) is a DF- space if and only if E is a DF-space.

Proof. Proceed exactly as in [12, 11.5 .11]. (Note that one on ly needs that S(p , K ) is ane- space an d an Soo-space.) 0

T he following lemma can be proved with essentially the same proof of [4, Thm. 4(b )]. Itest ablishes that the terms of a subscries summable sequence in S (/l , E ) must be supportedon the same set s.

Lemma. Le t ('Pn ) be a subseries summable sequence in S (/l , E ). Then, there exis t p air­wise disjoint set s A ), A2 , • • • , Ar E E, with p(A; ) > 0 for every i = 1,2, .. . , r , such thatfor all n E N we can write

'fin = L XA, zi(n ),i ",1

forsomezi(n) E E (i = 1,2 , ... ,n).

T he following is an extension of a result due to Bat t et al. [1].

T heorem 5. The scalar-valued space S(p , K) is not ultrabomoJogical. Hence neither isS(",E).

Proof. Assume, by contradiction , that S(p, K) is ultrabornological. We first show thatevery Banach disc in S(p , K) is finite-dimensional. Suppose, on the cont rar y, that there

296 Diaz et, ej, ,

exists a line arly independe nt seque nce (W..) in a certain Banach disc D of S(p,K). Foreach n E N, set I{Jn "" (n'2p V(tPn» - IIjIn , where PD is th e ga uge of D . T hen, t he sequence(IP n) is also linear ly independent and subseries sum mable in the Banach space spannedby D . Clearly, E lf'n is also subseries convergent in S(p,K) with the uniform convergencetopology. By the Lemma above, there exist pairwise disjoint sets A I , A2, . . . , A r E E, withIleA;) > 0 for every i = 1,2, .. . , r , such that for all nE N we can write

,'fJn = L XA;z;(n),

;=1

for some numbers z;(n) E K. It follows that the sequence (!f'n);:O=n. is linearly dependent.A contradiction .

Now, take any linear fonn f : S(I-I ,K) ..... K. All Banach discs in S(p,K) are finite­dimensional, hence f maps every Banach disc into a bounded set. Since S(p, K) is ultra­bomological, f is continuous. Thus, every linear form on 5(p, K) is cont inuous. But , aseasily seen, condit ion (e) implies that t he functional

, .If! = L XB;x; - L X;

; ""1 ;""1

is not conti nuous on S(p , K) . A contradiction.

Final ly, S(p , E ) is not ultrabomological because S(p, K ) is a complemented su bspaceof 5( p, E) = S(l l ,K ) 0 . E and t he property of being ultrabornological is inherited bycomplemente d subspac es [12, 6.2.21. 0

If we hav e a positi ve measure defined on a ring of sets, it m ay happen that the su bspaceof t hose classes of simple funct ions with finit e measure supp ort is strictly smaller than thewhole space S(p, E) . Let us consider a part icular case , namely the algebra p eN) of allsubsets of N with the cardinal measure, t hen the respect ive two spaces of scalar - valuedsimp le fun cti ons are widely known: The whole space is mo, the space of sequences withfinite ran ge, and the associated space of simple functions wit h finite support s is 4>, thespace of all sequences with only finitely many non-zero coordinates. Endowed with thesupremum nonn , q, is bom ological, but neither barrelled nor ultrabornological, while mois barrelled an d hornological but not ultrabornological. Theorem 5 above an d T heorem 6below say t ha t this behaviour is, in a sense, the best t hat can be expected.

Finally, we show t he difference between the behaviour of the whole space of simplefunct ions and t he subspace of functions with finite measure support (if t hey are not thesame, of course). So, suppose that there are sets in E wit h infinite measure. Then thefamily E I of all sets in E having finite measure is a prop er sebring of E and the restrictionPI of p to EI is also a positive finitely additive measure. If t here is a sequence of pairwisedisjoint sets with finite measure in E (the natural case where this happens is when p is apro per a - finite measure defined on a e - algebr a of sets ) then condit ion (e ) holds in Ef>and we can a pply Theorem 2 above to the space S (PI ' E). However , the situation is much

Diaz et al. 297

different concerning the applicability of Theorem 3 because, as we are about to see, if /-'is a proper e -finite measure defined on a c --elgebra of set s, then the scalar-valued space5 (/-'1 ' K) is not barrelled. Thi s will follow by applying our next result to this particularcase.

Theorem 6. Let /-' be a finite measure defined on a ring E of subsets of a set O. If /-' isunb ounded, then 5 (/-" K) is not barrelled. Hence neither is 5 (/-" E).

Proof. Let T be the unit ball of the L1-norm in 5 (/-" K), i.e.

We will show that T is a barrel in 5 (/-" K), but not a zero-neighbourhood.

First , we prove that T is a barrel. Obviously, T is absolute ly convex and absorbent .Let us see th at T is closed. Let (If..) be a sequence in T that converges to a functioncp E 5 (/-" K) for the sup norm. Let A be the support of cp. It is clear th at rp .. ' XA -+ 'P inthe L1-nonn. It follows easily that 'P E T .

To prove that T is not a zero-neighborhood, we shall see that T does not contain asuitable null sequence. Find a sequence (0..) in E such that /-,(On) > n . For n E N definerpn = Xo~/n. Clearl y, ('Pn) is a null sequence in 5 (/-" K). However , IIrp.. lb > I so thatrp nf/. T , for all n E N.

Alternative proof. Find a sequence (0..) of sets in E such that /-,( On) -+ 00 . For eachn E N consider the bounded measure /-' .. defined on E by /-' n(A) := p(A n On); notethat /-'n vanishes on all p-null sets. Then, the sequence (/-' n ) is set- wise bounded but notuniformly bounded, i.e . Nikodym's boundedness theorem fails for E/'Eo, and this showsthat 5(p , K) is not barrelled 0

Acknowledgement. Authors want to thank J. Bonet (Universidad Politecnica de Valencia ).His comments contributed to improve the readability of this paper.

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S . DLu, A. FERNANDEZ, M . FLOR ENCIO , P . J . PA UL: E . S . I NGEN I ER OS IN DUSTRIALE S, A VD A. REIN A

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