I ntrnat. J. Math. Math. Sci.
(1985)189-192
Vol. 6 No.
189
RESEARCH NOTES
DUAL CHARACTERIZATION OF THE DIEUDONNE-SCHWARTZ
THEOREM ON BOUNDED SETS
C. BOSCH, J. KUCERA, and K. McKENNON
Department of Pure and Applied Mathematics
Washington State University
Pullman, Washington 99164 U.S.A.
(Received August 3, 1982)
The Dieudonn-Schwartz Theorem on bounded sets in a strict inductive limit
ABSTRACT.
is investigated for non-strlct inductive limits.
Its validity is shown to be closely
connected with the problem of whether the projective limit of the strong duals is a
strong dual itself.
A counter-example is given to show that the Dieudonn6-Schwartz
Theorem is not in general valid for an inductive limit of a sequence of reflexive,
Frchet spaces.
Locally convex space, inductive and projective limit,
KEY WORDS AND PHRASES.
barrelled space, bounded set.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
Primary 46A12, Secondary 46A07.
INTRODUCTION
i.
This paper is written for those with at least an elementary knowledge of the
theory of locally convex spaces.
Let E
l
c E
2
c
n
U Em
[I].
be a sequence of locally convex, Hausdorff, linear topological
spaces such that each E
E
A good reference is the book of Schaeffer
is continuously contained in
En+l,
is Hausdorff as a locally convex inductive limit.
and such that the union
It is obvious that any
m i
bounded subset of a space En
is
also bounded in E.
arises in this way, we shall say that the DSP
If each bounded subset of E
(Dieudonn-Schwartz Property) holds.
A
well-known theorem of Dieudonn-Schwartz states that the DSP holds provided each En is
closed in
En+ I
and has the topology inherited from
En+ 1
(see [i] or [4] 11.6.5).
C. BOSCH, J. KUCERA AND K. MCKENNON
190
c E
In duality theory an increasing sequence E
ing sequence F
F
1D
c
2
corresponds to a decreas-
where each F is dual to E
n
n
2
m=l
endowed with the projective limit topology induced from the weak topologies
may be identified with the dual of E relative to the weak topology o(F,E)
In applications the strong topologies
(Fn,En)
projective limit (F) induced by these on F.
least as fine as (F).
F
The intersection F
m
(Fn,En),
([I] IV.4.5).
are often of interest, along with the
The strong topology B(F,E) is always at
The problem of determining when (F) equals 8(F,E) turns out
A precise
to be closely connected with determination of the validity of the DSP.
statement of this is given in the theorem below.
THE QIU PROPERTY
2.
Recent work by Qiu
[2]
Let B
suggests a slight relaxation of the DSP.
set of all subsets B of E such that B is bounded in some E
be the
We say that the QP (Qiu
n
B.
Property) holds if each bounded subset of E is contained in the closure of some B
For spaces V and W dual to one another, and a subset S of V, we write
o(V,W)-closure of S
in V and S
W
V
for the
for the polar of S in W.
convex hull of S, the Bipolar Theorem
([I]
Thus, if <S> denotes the
--V
IV.I.5) states that <S>
(S W oV We
note for use below that the polars of the closed, radial, convex bounded subsets of V
are just the barrels of W, and vice versa.
A necessary and sufficient condition for the QP to hold is that
THEOREM.
8(F,E).
(F)
Then B
set of E.
D
F c B
F
F
is a barrel and so contains a
8(F,E) now follows that there
From (F)
A
8(F,E), and let B be an arbitrary bounded sub-
Suppose first that z(F)
PROOF.
F.
ts just D
AoEn
Letting D
Fn
A
F
(BF)E c (DoF) E.
8(F,E)-open neighborhood of 0.
is a barrel
A in some F
we see that D is bounded in E
F, we have D oF c B
oF,
such that
F
A.
and D n
n
n
Consequently, B c
Since the Bipolar Theorem guarantees that
Because
<B->
(DoF) E
is just the
closure of D in E, we have shown that the QP holds.
Now suppose that the QP holds, and let A be an arbitrary barrel of F.
is bounded and so there exists a bounded set B of some E
closure
- =B’"
a bounded
of B in E.
set),
Since
BF
it follows that B
B
F
Fn
F and B
Fn
n
Then A
E
such that A E is in the
is a barrel (being the polar of
is a z(F)-neighborhood of
O.
But we have
DIEUDONN-SCHWARTZ
AE c
<--E
E c
191
so
BoF (-ffE)
We have shown that 8(F,E) c z(F).
3.
THEOREM ON BOUNDED SETS
F c
(A
oE) oF
A.
Q.E .D.
The reverse inequality is evident.
COUNTER-EXAMPLE
[3]
It was demonstrated in
that the DSP holds when all the E
The following example shows that, for reflexive
Banach spaces.
are reflexive
n
Frchet spaces,
even
the QP may fail to hold.
For each n
lq, let D
{1,2
be the regionIR
n
n} and let E be the linear
n
For n,m
space of functions infinitely differentiable on Dn
pact set
{x
Ixl
D
n
Ix-
< m,
> I for all j
J
n} and, for each f
1,2
m
be the comlet K
n,m
lq
E
n
let
IIl..f.
Then each E
{I[
[In, m
},
0 1
m}
0
i
n,m
E
m=l
Frchet space ([4] III 8 3).
is a nuclear
([4] 111.7.2,
a Montel space
U
K
x
equipped with the locally convex topology generated by the family
n
m
that E
sup{mlf (i)(x)l:
,m
n--
n
En.
x
(x
IR, let f n (x)
lg, and x
sup{If n (i):
Dn[n
+ I]
i, n
is contained by
V..Evidently
2
n)
0
i
..,n
e
2
-(x-n)
I}
and let
Clearly, each f
Inc fn
is in W for n
m
n
is in
II lll,m -unit
Then, for some m lg, the
Let V be any neighborhood of 0 in E.
ball W of E
n
We proceed to show
Corollary 2) and thus reflexive.
does not have the QP.
n
For each n
c
Hence each E
+ I,
m
+ 2
n
Consequently there exists k > 0 such that
then set B
(h
n
n
h
n
nc
f
n
n
is finite.
lqmthat
is,
lq} is bounded in E.
Let D be a bounded subset of one of the spaces E n
M
kV for all n
sup{lh (n+l)(x)l:
-
x
[n +
,
Then the number
3
h
+ ],
n
D}
(3.1)
Let p be the polynomial (with non-vanishing constant term) such that
_i
h(n+l)
(x)
n+l
(x
n
I)
-(x-n-l)
p(x) e
2
for all x
Dn+ I.
Evidently there exists some r > 2 such that
inf
h(n+l)
I..n+
I
(x)
l:
x 6
[n +
,
n
+
> M
+ 2.
(3.2)
192
C. BOSCH, J. KUCERA AND K. MCKENNON
r and n + i.
Let K be any integer larger than
II ..llm,k -unit
ball of
En.
Then, for each g
sup{Ig (n+l) (x) l:
In
x
For each m
Sin,
,
+
lq, let
Sm be
the
we have
k]}
+ 1
n
(3.3)
< i.
Evidently this last inequality also holds for all g in the convex hull H of the union
u S.
m=l
From (3.2) and (3.3) follows that the set
E such that, for each g 6
inf{
hn+I
+ H
hn+1
is a neighborhood of
hn+I
in
+ H,
g(n+l )(x)l:
[n +
x
Hence, (3.1) and (3.4) imply that D o
1
-,
(hn+I
+ H)
n
.
+
k]}
> M + I.
(3.4)
Thus the QP does not hold.
REFERENCES
I.
Schaefer, H. H.
Topological Vector Spaces, McMillian, New York, 1966.
2.
Qiu, Jing Huei.
Some results on bounded sets in inductive limits (to appear).
3.
Kucera, J., and McKennon, K.
Dieudonn-Schwartz
inductive limits, Proc. Amer. Math. Soc. 78
4.
Dieudonn6, J., and Schwartz, L.
Theorem on bounded sets in
(1980), 366-368.
La dualit dans les espaces (F) et (F), Ann.
Inst. Fourier (Grenoble) i. (1942), 61-101.