Indag. Ma he n., N.S., 5 (l), 51-60
KG he echelon spaces CI la Dieudonnk
Ma ch 28, 1994
by Miguel Flo encio, Ped o J. Paljl and Ca men Shez
Depa amen o de Ma emci ica Aplicada II. E.S. Ingenie os Indus iales, A . Reina Me cedes s/n,
41012-Se illa, Spain
Communica ed by P o . J. Ko e aa a he mee ing o Decembe 21,1992
ABSTRACT
Le (gn) be a sequence o locally in eg able unc ions de ined on a Radon measu e space. The
echelon space associa ed o (g,,) was de ined by J. Dieudonne as he Ko he-dual o (g”). i.e. he
space n o all locally in eg able unc ions such ha all he in eg als s I g,l a e ini e. Deno e
by Ax he Kii he-dual o A. We p o e ha A@(,$ Ax)) IS a F eche space wi h dual Ax. This e-
sul gi es i s co ec sense o a w ong a i ma ion o J. Dieudonne and alida es hose ins ances
whe e i has been used. As a ool o p o e his esul , we s udy he p oblem o when he s ong
dual o a pe ec space coincides wi h i s Kii he-dual and gi e some necessa y and su icien con-
di ions.
1. INTRODUCTION
The Ko he-Toepli z heo y o pe ec sequence spaces [7, $301 has been one o
he mos in luen ial in he s udy o he s uc u e o locally con ex spaces. This
heo y, and in pa icula he duali y be ween echelon and co-echelon sequence
spaces (see e.g. [l] and [6]), has p o ided he specialis s wi h plen y o hin s, ex-
amples and coun e examples. Di e en ex ensions o his heo y a e ob ained by
eplacing he C in he de ini ion o G. K&he and 0. Toepli z wi h a sui able S.
One o he i s mo es in his di ec ion was made by J. Dieudonne.
I is in he las sec ion o his seminal pape [3, $161 ha J. Dieudonne conside s
echelon spaces o unc ions. His de ini ion is as ollows: le X be a locally compac
and a-compac Hausdo opological space wi h a Radon measu e p. The eche-
lon space associa ed o a sequence (g,) o locally in eg able unc ions is de ined as
he Ko he-dual o he sequence:
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n := : is locally in eg able and
&n( ) = s I .g,I dp < +oo, n = 1,2,. . .
x
A e gi ing his de ini ion, he a i ms ha A endowed wi h he opology de ined
by he semino ms ( pg,) is a F eche space and ha i s opological dual coincides
wi h i s Ko he-dual Ax. This is no co ec , as was poin ed ou by J.A. Lopez
Molina [8, Ex. (p. 187)] wi h he ollowing example: conside he uni in e al wi h
i s Lebesgue measu e and he echelon space A associa ed o he unc ion
g(x) = exp( - l/x). Then h = x[~,~I is a unc ion in ilx because he unc ions in A
a e, by de ini ion, locally in eg able. Howe e , h is no con inuous o he semi-
no mp, because he sequence (k xp~,k]) om n sa is ies
lip pg(k. XIO,llk]) = lip s k. exp(-l/x) dx 5 lip exp(-k) = 0,
0
bu o allk= 1,2,...weha e
jkqqo,,,k,(x).h(x)dx= ykdx= 1.
0 0
This mis ake led J.A. Lopez Molina o conside an al e na i e de ini ion o
echelon space: equi e he unc ions in n o be only measu able ins ead o locally
in eg able. The heo y o echelon spaces de ined in his way has been de eloped
and gene alized by J.A. Lopez Molina [8] and [9], J.C. Diaz [2] and K. Reihe [12].
Ou pu pose in his pape is o p o e ha J. Dieudonne’s a i ma ion is essen-
ially co ec in he sense ha an echelon space A is a F eche space when en-
dowed wi h he s ong opology p(A, Ax) and i s opological dual equals i s
Ko he-dual Ax. This esul will, in u n, alida e hose ins ances in which
J. Dieudonne’s a i ma ion has been used p ecisely in i s co ec sense, as in [ll,
Co . 11.
In $2 we ecall J. Dieudonne’s de ini ion o pe ec spaces and gi e some
necessa y and su icien condi ions o he opological dual o a pe ec space o
coincide wi h i s Ko he-dual. These condi ions, which a e o independen in e es ,
gene alize and uni y some p e iously known esul s and will be used in $3, whe e
he esul announced in he p eceding pa ag aph appea s.
We e e he eade o W. Rudin’s book [13] o he esul s conce ning measu e
heo y and in eg a ion, and o G. Ko he’s monog aph [7] o he heo y o locally
con ex spaces.
2. WHEN DOES THE TOPOLOGICAL DUAL COINCIDE WITH THE KGTHE-DUAL?
Al hough some o ou esul s in his pape can be gi en in a mo e abs ac
measu e- heo e ic ame, we shall s ick o J. Dieudonne’s o iginal o mula ion
[3, #lo-161. In wha ollows, X s ands o a locally compac , Hausdo opolo-
gical space ha is a-compac , so ha we can w i e X = U, X,, whe e e e y X, is
52
compac and X, c in (X,+i) o all m E N. Le p be a posi i e Radon measu e
on X and G be he space o all (equi alence classes o ) locally in eg able unc ions
om X in o he ield [ib o eal o complex numbe s. Fo a subse A c R we deno e
by A” he se o all g E $2 such ha . g is in eg able o each E A; Ax is called
he K&he-dual o A. A linea subspace A o 0 is said o be a pe ec space (o
Kii he space) i (Ax)” = A. In pa icula , Ax is always pe ec . Fo ins ance,
L1 (,u) and L”(p) a e pe ec spaces, and each one is he Ko he-dual o he o he .
The space l is also pe ec and Qx is he space @ o all (equi alence classes o )
measu able essen ially bounded unc ions wi h compac suppo . When A con-
ains @, he spaces A and Ax a e pu in o duali y by means o he canonical bi-
linea o m
( > g) = L (x) g(x) 44x) ( E 4 g E Ax 1.
Le B s and o he uni ball o L”(p). A subse H o R is called no mal i
E H o all E H and E B o , equi alen ly, i E H and g is a measu able
unc ion such ha /g(x) 1 5 1 (x) 1 ,+a.e. on X, hen g is also in H. The no mal hull
o a se H c 0 is de ined by { : E H, E B}. One impo an ac is ha he
no mal hull o a weakly bounded se is also bounded [3, P op. 61. This means ha
he s ong opology /?(A, A”) is gene a ed by he semino ms
whe e H uns h ough he absolu ely con ex, no mal and o(A’, A)-bounded
subse s o Ax. I A is a pe ec space, hen A(p(A, Ax)) is comple e [3, Th. 51.
Deno e by A’ he opological dual o A(p(A, Ax)). The p oblem o when A’
equals Ax o , equi alen ly, when he s ong opology p(A, Ax) coincides wi h he
Mackey opology ,u(A, A”), has been add essed by se e al au ho s (we shall gi e
p ecise e e ences in he no es ollowing ou Theo em l), and di e en necessa y
o su icien condi ions ha e been conside ed. In ou i s esul we uni y and ex-
end p e iously known esul s.
Theo em 1. Le A be a pe ec space and deno e by A’ he opological dual o A en-
dowed wi h he s ong opology /?(A, A”). C onside he ollowing condi ions..
(i) A’ = Ax o , equi alen ly, p(A, Ax) = P(A, Ax).
(ii) Fo e e y unc ion E A and e e y sequence (A,) o measu able se s such
ha lim, p(A,) = 0, he sequences ( XA,) and ( - . xx,) con e ge o ze o o
he s ong opology P(A, Ax).
(iii) I ( n) is a dec easing sequence in A ha con e ges o ze o p-a.e., hen ( n)
con e ges o ze o o he s ong opology ,@A, Ax).
(i ) A(p(A, Ax)) is sepa able.
( ) The space Cc(X ) o con inuous unc ions wi h compac suppo is dense in A
o he s ong opology p(A, A”).
53
Then we ha e (i ) + (i) @ (ii) @ (iii) + ( ). Besides, i he measu e space
(X, p) is sepa able, hen (i ) is equi alen o (i) - (iii). Mo eo e , cyX is a me iz-
able space, hen all he condi ions abo e a e equi alen .
P oo . We s a by p o ing he equi alence o (i) - (iii).
(i) + (ii): Take E A. T o see ha ( - XX,,) con e ges s ongly o ze o, ix a
s ong semino m pH, whe e H is an absolu ely con ex, no mal and a(A’, A)-
bounded subse o Ax. Condi ion (i) ells us ha H is ela i ely compac o he
weak opology a(Ax , _4). Conside he mapping
T:gEAx-T(g)= .gEL’(p).
Since A is no mal, he unc ion . h is in A o e e y h E L”( p), and we ha e
(T(g),h)(,l(,),,,(,)) = i .g.hd~ = (gJ.h)(,x,,).
The e o e, T is a(A’, A) - a(L’( p), L”(p)) con inuous and consequen ly T(H)
is ela i ely compac o he weak opology a(L’( p), I.,“( p)). Then, by [3, Th. 41,
he e is a compac se K c X such ha
sup J” IT(g)]dp:gEH J” j .gldp:gEH
J’ K X K
TakingmENsuch ha KcX,weha epH( - .xxm)<l o n>m.
On he o he hand, i p(A,) + 0 bu ( . XA.) does no end o ze o o he
Mackey opology, hen we can ind a no mal se H c Ax, absolu ely con ex and
a(Ax , A)-compac , and an inc easing sequence o indices (nk) such ha
bu
sup Aj 1 .gldl.L: gEH >l o allk=1,2,...
“k
Conside he dec easing sequence (Fk) o measu able se s de ined by Fk :=
Ujzk A,. Then P(h) 5 xj>k P(&), h ence limk p(Fk) = 0. Fo each k E N
de ine he se
H/c:= gEH: j- I .gldp> 1
Fk >
These se s (Hk) a e non-emp y, o(A" , A)-closed and hey ha e he ini e in e sec-
ion p ope y because o kl < k2 we ha e
J 2 I Wd~ I .hldp
I
o e e y h E Ax so ha HkZ c Hk,. Since H is o(A’, A)-compac , he e is a
unc ion g E H such ha g E Hk o all k o , equi alen ly,
J I . gl dp 2 1 o all k E N.
Fk
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Bu . g E L’(p). The e o e, he e is some 6 > 0 such ha i p(A) < 6 hen
J, I . gl dp < 1, and his is in con adic ion wi h he inequali y abo e because
limk p(Fk) = 0.
(ii) + (iii): Le ( n) b e a d ec easing sequence in A ha con e ges o ze o p-a.e.
and ake a no mal and a(nx, A)-bounded se H. Applying (ii) o = i E A we
can ind an index N E N such ha
Now, le cy =PH(xx,,,). We may assume ha cy # 0. (O he wise, e e y g E H
would be ze o /l-a.e. in X, and he p oo o his implica ion would be inished.)
Conside he measu able se s de ined o each II E N by:
A, := {x E -=,: n(x) > 1/(4d’)}.
Since p(XN) < +oc and ( n) con e ges o ze o p-a.e., we ha e ha lim, CL(&) = 0.
By condi ion (ii), he e is m E N such ha ~~( i . XA,) < $. Now, o n 2 m we
ha e ha I n(x)I < 1/(4a) on XN &.The e o e
PH(h XX,) < PH(h . XA,) +PH( n ’ % A,)
5 PH( i XA,,,) + (l/(4a)) ‘PHh. A,)
< ; + (1/(4a)). cy = 1.
Hence, i n > m we ha e
pH( n) 5 pH( n - n XX,) +pH( n XX,) < ; + ; = 1.
(iii) =+ (i): W e a ways ha e Ax c A’, On he o he hand, ake 4 E A’. Fix an
1
index N E N. Fo each measu able se A c XN, de ine G(A) = I. Condi ion
(iii) yields ha G is a a-addi i e measu e. Indeed, i (A,) is a sequence o disjoin
measu able se s in X, wi h union A, apply (iii) o he unc ions de ined by
, := XA - 2 X.4, E @ c A, o eachm= 1,2,...
n=l
Then ( m) con e ges o ze o o he s ong opology. Hence I = C,“=i ~(xA,)
so ha G(A) = C,“=, G(A,). Le us see now ha G is absolu ely con inuous wi h
espec o p: i we ha e a measu able se A C XN wi h p(A) = 0, hen XA = 0 p-
a.e., hus G(A) = I = 0.
Apply he Radon-Nikodym Theo em o deduce he exis ence, on each XN, o a
unc ion gN E L’(p, X ) such ha
G(A) = JgN dp o A c x .
A
I is clea ha i N 2 M, hen gN = gM on X, so ha he unc ion g = limN gN is
well-de ined and locally in eg able. Mo eo e , o each compac K and each
measu able se A c K, we ha e
G(A) = s g dl.L.
A
55

We p o e now ha g E Ax and 4( ) = ( , g). Suppose, wi hou loss o gene ali y,
ha g > 0. We p oceed in se e al s eps.
(1) Since 4 is linea , we ha e ha 4( ) = (g, ) o a simple unc ion wi h
compac suppo .
(2) Fo E @ ( he space o measu able and essen ially bounded unc ions wi h
compac suppo ), le (&) b e a ne o simple unc ions wi h compac suppo such
ha lim, Il( - u) XA llm = 0 w h e e A is he suppo o . I H is a no mal and
a(Ax, A)-bounded se , hen we ha e
limPH( - o?) = lim sup
a J I - ol] . IhI dp: h E H >
~~11~ .PH(XA) = 0.
N
This p o es ha ( ol) con e ges o o he s ong opology p(A, Ax). Using his,
(1) abo e and ha = lim, a in Lm(p, A), we ob ain
(3) Now ake a posi i e unc ion E A. Fo each 12 E N, le n be he unc ion
in @ de ined by:
h(x):= {y i (x)<nandxEX,
> o he wise.
Then ( - n) is a dec easing sequence ha con e ges o ze o p-a.e. By condi ion
(iii), n con e ges o o he opology @(A, A”). Using his and he Mono one
Con e gence Theo em in L’ (p) we ha e
(4) Finally, o a bi a y E A he equali y 4( ) = ( , g) ollows by linea i y.
(i ) + (i): W e a wa s
1 y h a e Ax c A’. Now, ake 4 E A’. Then he e is some ab-
solu ely con ex, closed and o(Ax, A)-bounded subse H o Ax such ha 4 E H”“,
he bipola o H in A’. Since A(p(A, A”)) is sepa able, H”” is me izable
[7, §21.3.(4)] and compac o he opology o(A’, A). The e o e, H is sequen ially
dense in H”“. Bu , on he o he hand, H is o(Ax, A)-sequen ially comple e
because i is closed in A’(c~(il’, A)) and his space is sequen ially comple e
[3, P op. 121. Hence, c5 E H”” = H c Ax. Consequen ly, A’ c Ax.
(i) + ( ): Fo e e y non-ze o g E Ax he e is a unc ion h con inuous and
ha ing compac suppo such ha (h, g) # 0 [3, p. 981. Then, by he Hahn-Banach
sepa a ion heo em, Cc(X) is dense in A o any opology such ha he dual o A
isAx.
(i) - (iii) + (i ) when he measu e space (X, p) is sepa able: Fo e e y non-
ze o g E A” he e is a simple unc ion h ha ing compac suppo such ha
(h,g) # 0. Condi ion (i) and he Hahn-Banach sepa a ion heo em ensu e ha
he space SC(X) o simple unc ions ha ing compac suppo is dense in
56
n(,B(A, A”)). The e o e, we ha e o p o e ha S,(X) is sepa able o he s ong
opology /?(A, A”).
Since he measu e space (X, p) is sepa able, he e is a coun able amily C o
measu able se s (we may, and do, assume ha his amily con ains all he se s o
he o m Y ~ X, o Y E C and n E N) such ha o e e y measu able se A he e
is a sequence ( Yj) om C wi h limi p(AnYj) = 0, whe e D s ands o he sym-
me ic di e ence ope a o . I , in pa icula , A is con ained in X,,, hen bo h XA
and xX, A a e in SC(X) c A. Since limj p(An Y,) = 0, we can apply condi ion (ii)
o xA andxx, A = X& - XA ( ecall ha A C X,) o ob ain, o he s ong opol-
ogy P(4 AX )9
O=limxAx~a ,=limxA(xA+x~-2xAx~)=li~(xA-xAxY;)
.I I
and
0 = lim(xx;, -xA)xAa , =lijn(x~ -XA)(XA +,YY,-~xAxY,)
i
= Ii,? k,~x , - x~xY,).
I ollows ha limj (XA - x~~,~~,)) = 0 o he s ong opology p(il, Ax). This
ensu es ha he coun able se D o all simple unc ions o he o m C c yxy,
whe e each Y in he sum is om C and each QY is a ional, is dense in S,(X) o
he s ong opology /?(A, /lx).
( ) =+ (i ) when X IS me izable: Acco ding o condi ion ( ), we ha e o p o e
ha Cc(X) endowed wi h he es ic ion o he opology @(il, A”) is sepa able.
Now, i X is me izable, hen each o he spaces C(Xn)( 1) 11,) is sepa able. Fo
e e y n E FU, le D, be a coun able dense se in C(Xn)( j II,). Now, o E C(Xn)
ake g E D, such ha
I (x) -g(x) I 1 o all x E X,.
Le H be a no mal and a(n”, A)-bounded subse o Ax. Since S and g a e sup-
po ed in X,, we ha e
~~( ‘-g) = SUP . I -s/. Ihldp: h E H
xn
This shows ha D = (J, D, is a coun able se dense in Cc(X) o he s ong op-
ology p(n, A);). q
No es. J. Dieudonni: [3, $13 (p. 107)J claimed ha (i ) =+ (i). Y. K6mu a [5,
Th. 1.31 showed he equi alence o (i) and (i ) o X = [w”. G.G. Lo en z 110, Th. 31
p o ed ha (iii) + (i) o he case when X is a ini e in e al in he eal line and A
is a no med space. R. Welland [18, Th. 21 p o ed he equi alence o (i), (iii) and (i )
unde he hypo hesis ha he measu e space (X, ,u) is sepa able; he e we ha e
shown ha his hypo hesis is no eally necessa y o p o e (i) ej (iii). Condi ion
(ii) may be easie o use ha (iii) and he p oo o (ii) + (iii) abo e ollows he
57
ideas gi en by A.C. Zaanen in [19,§72]. Finally, condi ion ( ) has been conside ed
by G. Sil e man o ansla ion in a ian pe ec spaces [17, Th. 2 and Th. 31.
3. THE STRONG DUAL OF AN ECHELON SPACE
Le (g,) be an inc easing sequence o non-nega i e locally in eg able unc ions.
The Ko he-dual o (gn) is called he echelon space associa ed o his sequence and
we shall deno e i by A(g,). Thus
A(a) = E Q: ~g,( ) := j- I (x)1 .8,(x) dp(x) < +cc
x
o alln=1,2,...
When (g,) educes o a simple unc ion g, he space A(g) is commonly deno ed
by Li and has many in e es ing p ope ies (see [3, #IO-121, [4] and [15]). No e ha
we can w i e A(g,) = n, Li.. As we said in he in oduc ion, he main pu pose o
his pape is o gi e he ollowing esul .
Theo em 2. Le A = A(g,,) be he echelon space associa ed o an inc easing se-
quence (g,) o non-nega i e locally in eg able unc ions. Then A is a pe ec space
wi h Kii he-dual Ax = U, (Lin)‘. Mo eo e , A(p(A, Ax)) is a F che space wi h
dual Ax and he s ong opology ,B(A, A”) is gene a ed by he amily o semino ms
pg,: ~A+p,~( ) :=EI (x)l.g.(x)d~(x), n= 1,2,...
qm: 6 A-+ qm( ) :=L I (x)Idp(x), m= I,&...
P oo . Fi s , we p o e he heo em when (gn) educes o a single unc ion g E 6).
Acco ding o [ 15, P op. 11, he s ong opology ,B( Li, (Li) ’ ) is gi en by he amily
o semino ms { pg and qm, m = 1,2, . .} whe e pg( ) = Jx ) ) g dp. The space
Lk is pe ec because i is he Ko he-dual o {g} and he e o e, i is comple e
when endowed wi h he s ong opology [3, Thm. 51. Hence Lh(p(Li, (Lh)“))
is a F eche space. To p o e ha he opological dual o his space is (Li)‘, we
shall apply condi ion (ii) o Theo em 1 abo e. Le E Lj. Ob iously,
lim, qm( - . xx”) = 0 o e e y m E N, and since . g E L’(n), we also ha e
lim pg( - XX.) = lip s 1 gl dp = 0.
n x xl
Now le (An) be a sequence o measu able se s wi h lim, p(A,) = 0. Since
XX, is in eg able o each m, we ha e
li,m qm( . x.4,) = lip j I I xx, dp = 0.
A.
58
On he o he hand, using again ha g E L’(p), we ha e
l$p,( . XA,) = 1,” l I I . gdp = 0.
A”
This inishes he p oo o LL.
We u n now o he gene al case. Le T be he opology de ined by bo h se s o
semino ms { ps, : II = 1,2, . . .} and {qm : m = 1,2, .}. Since n c Ljn o e e y
n E N we ha e ha (Lin)’ c Ax and he e o e U, (LL”)” c Ax. Deno e by ,, he
co esponding s ong opology ,B(Li., (Li”)’ ) on Lj.. Since g, < g,+l, he inclu-
sion $+, (7,+1) + Lin (~~1 IS con inuous, so ha A(T) is he educed (because
Cc(X) is dense in each o hese spaces by (i) + ( ) in Theo em 1) coun able p o-
jec i e limi o a amily o F eche spaces. The e o e, .4(~) is a F eche space.
Deno e by A’ he dual o n( ). No e ha T = @(A, A’). On he o he hand,
A’ c u (L;n(Tn))’ = u (L;“y c AX.
” n
The p oo will be inished i we show ha il” c A’. Take h E Ax and assume,
wi hou loss o gene ali y, ha h 2 0. Call T he linea o m induced by h on il,
T( ) = ( , h). Fo k = 1,2,. . de ine he unc ions
h/‘(X) = i
h(x), i h(x) < k and x E xk
0, o he wise,
hen (hk) is an inc easing sequence ha con e ges poin wise o h. We can apply
he Mono one Con e gence Theo em o deduce ha
T( ) = S .hd,u=lip ~ ~hkd~=li~( ,hk).
x x
Now, obse e ha e e y hk E @ and ha he semino m I(., hk) 1 is domina ed by
k qk. The e o e, T is he poin wise limi o he sequence ((., hk)) o -con inuous
linea o ms. By he Banach-S einhaus Theo em, T is also -con inuous, i.e.
hEA’. q
One can see now ha he ouble in J. Dieudonne’s a i ma ion was ha he
missed he amily o semino ms {qnl : m E N}.
ACKNOWLEDGMENT
This esea ch has been suppo ed by La Conseje ia de Educa ion y Ciencia de
la Jun a de Andalucia.
REFERENCES
1. Bie s ed , K.D., R. M&e and W. H. Summe s ~ Kii he se s and Kii he sequence spaces. Func-
ional Analysis, Holomo phy and App oxima ion Theo y, No h-Holland Ma hema ical
S udies 71, Else ie iNo h-Holland, Ams e dam, Ox o d and New Yo k, 27791 (1982).
2. Diaz, J.C. -S ic ly egula echelon Kii he spaces. A ch. Ma h. (Basel) 46,360-367 (1986).
3. Dieudonne, J. - Su les espaces de Ka he. J. Analyse Ma h. 1,81-115 (1951).
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