Internat. J. Math. & Math Sci.
Vol. 8 No. 4 (1985) 813-816
813
RESEARCH NOTES
THE SPACES Oa AND Oc ARE ULTRABORNOLOGICAL
A NEW PROOF
JAN KUCERA
Department of Mathematics
Washington State University
Pullman, Washington 99164-2930, USA
(Received February 28, 1985
ABSTRACT.
[1] Laurent
In
@
Schwartz introduced the spaces 0 and
of multiplication
M
Then in [2] Alexandre
and convolution operators on temperate distributions.
0
Grothendieck used tensor products to prove that both
and
are bornological.
proof of this property is more constructive and based on duality.
Our
Temperate distribution, multiplication and convolution,
inductive and projective limit, bornological, reflexive, and Schwartz spaces.
1980 MATHE.fATICS SUBJECT CLASSIFICATION CODE: Primary 46F10, secondary 46A09.
KEY WORDS AND PHRASES.
We use C, N, R, and Z, resp., for the set of all complex, nonnegative integer,
For each q
N, the space
real, and integer numbers.
{f: Rn
eq
C;
2
llflq
x2a IDBf (x) 12
fn
l
I+I <q
R
dx <
+ oo}
is Hilbert.
We denote by L
Here Df stands for the Sobolev generalized derivative.
II’II
dual of L and by
the standard norm on L
Then the space S
-q
q
-q
decreasing functions, resp. its strong dual S’
is the proj lim L
q
q
ind lim L
-q
q
T
k
Lm
ind lim
p+=
N
q
0
M
wkf:
f
k
Z, the image of
k, m
WLm
wPL q
Lm under
5’
T
k
and
the space
-q
=@n
is reflexive and
The strong dual
N.
Further,
for its strong dual.
The space
for each p
S’
@_q
@q,
n
R
of
@M equals
q’ q
N.
[7]
w-PL
equals proj lim
q
@M
N, stands for the
q
It is proved in
proj lim
is dense in each L
Then
We
is injective.
Ix12) 1/2 x
denote by WKL-m’
and provide it with the topology which makes
of multiplication operators on
PROOF.
Z
k
a topological isomorphism.
PROPOSITION
wPL q
S’
resp.
(I +
It is convenient to introduce the weight-function W(x)
The mapping T
the strong
-q
of rapidly
0
q
Finally, the space
see
ind lim
q
Hence
and $ fortiori its superset
that for each
[6].
O_q.
wP
@M’
is dense in
are dense in each
J. KUCERA
814
Oq
wPLq
indlim
p/
[3,
By
N.
q
4.4],
ch. IV,
OM’
the dual of
equipped with the
The Mackey and strong topologies on
Mackey topology, equals ind lim
q
is semireflexive,
as a projective limit of reflexive spaces
coincide since
@_q.
q,
M
[3,
see
5.5].
ch. IV,
O_q.
is the strong dual of ind lim
PROPOSITION 2.
q
By [3
PROOF
with the duality <
0M’ @
B(
@M’ M >
LEMMA I.
@M is
@
N.
q
is finer than
that
c L
q
and every set bounded in
q
W-rL q
and closed, set in
bounded
Let B be an absolutely convex
WrL -q
Then
By [3, Ch. IV, II.I,
is weakly sequentially compact and every sequence in B contains a subse-
2], B
{fk
quence
is consistent
q
is the strong dual.
W-rL q
Then
B is weakly compact as a polar of a neighborhood in
Cor
@
q/
[5, Prop. 4]
reflexive and
+ [1/2 hi,
is relatively compact in L
PROOF
proj lira
Hence T is coarser than the strong topology
The space
Let r
@M
the topology T of
On the other hand, it is proved in
)"
THEOREM 1.
W-rL q
4.5]
ch. IV
Since the set
{wr+qf;
B}
f
is bounded in
L2(Rn),
n, II
LI(R
O.
We may assume g
B.
which converges weakly to some g
{wqf;
the set
B}
f
is
{FP;
f
B}, where Ff is the
q, the set
N
n) and for any
Hence
on R
equicontinuous
locally
and
bounded
is
uniformly
of
f,
transform
Fourier
bounded in
{fk
n.
{fk },
contains a subsequence, let it be again
n
q.
N
for all
n, lal
uniformly on R
such that {D
Ffk(x)}
converges
inh(ix), iN.
and put
h(x)dx
n
>
O, there is
e
Given
B.
on
uniformly
of
L
f as i
in the topology
h
Then f
q
g for any f
B N
B. We fix this i. For every
N such that
i
q <
n
converges uniformly to 0 on R as k
q, the sequence
+
hi(x)
Take a non-negative function h @
,
i
llf-f,hill
Is 81
,
{WD(Ffk
and has an integrable majorant from
fk*
h
for
Fhi)}
Hence F(f
2.
0
LEMMA 2
W-rL-q
Let r
+ [1/2n
q
h
O, and
i
N.
Then
W-rL -q
e L
fortiori
N so that
If we choose k 0
both in the topology of L q
> k
then
If k q < 2E for k > k O.
O,
i
,
k
-q
n,
Ill k,hiIIq
< E
and every set bounded in
is relatively compact in L
PROOF
-q
Let B be an absolutely convex, bounded, and closed
set in
the same argument as in Lemma I, every sequence in B has a subsequence
converges weakly to some g
B.
W-rL
By
which
{fk
O.
We again assume g
wrLq.
W-rL
WrL
Let A be the
Denote by If"
the norm in
resp. Ii’ll
resp.
r,q
-r,-q
-q
B}.
f
and a
B the open unit ball in
closed unit ball in L
sup{Ilfll
0
r,-q
q
q
Since Lq is dense
Choose g > O. By Lemma
A is compact in the topology of
WrL q
in
For any
I<
WrL q
there exists a finite set
A, there exists
i
{i"
such that
F}
i
c L
q
ll-illr,q
such that A
<
g
’ fk >I I< -i fk >I + l<i fk>I II-ilr,q Ilfkl-r,-q
ga +
I<i, fk>l. If we choose k0 N so that l<i, fk>I < e for
k > k
0
and the sequence
{fk
converges in L
-q
U{@i
+
and for any k
B0;
i’
all i
i
N we have
fk <
F and
F}.
815
ULTRABORNOLOGICAL SPACES
By Lemma 2, for every p
PROOF.
r
+ [1/2n
is compact in
@
proj lim
w-PL -q
-q
k
Ek+
(1)
is a Schwartz space.
N the closed unit ball is
w-PL
By [4
Prop. 9]
Ch. 3.15
c
2
N, continuous and E
c E
where
the space
be locally convex spaces with identity maps:
Assume:
Hausdorff.
k
k+
every set bounded in E is bounded in some Ek,
I, k
w-r-PL-q
is Schwartz.
Let E
PROPOSITION 4.
E
N, 0 -q
For each q
PROPOSITION 3.
ind lim E
(2) every E k is a Schwartz space.
Then E is a Schwartz space.
[4, Ch. 3.15]
[4].
Proposition 4 is slightly more general than Prop. 8 in
proof requires only minor changes of the proof presented in
@
THEOREM 2.
@
Further,
is a Schwartz space.
ind lim @
@M
We have
PROOF.
q+
Each space @
-q
Fr{chet.
is Schwartz and
-q
is reflexive, hence quasi-complete, which in turn implies fast complete-
By [8, Th. 1], the assumption (I) of Prop. 4 is satisfied and
hess.
and its
@M
is a Schwartz
space.
@M
THEOREM 3.
PROOF.
duced in
which f
is complete.
The space
[I].
B of C
[--wmf
equals indlim
m
see
wm
m
by
4.4].
Ch. 2
m
Also,
Then the strong dual
@C
is isomorphic to
Hence it suffices to prove that ind lim
transformation.
Let F be a Cauchy filter on
the filter with base
C’
was intro-
and provide it with the topology for
is a topological isomorphism.
[2
&
functions, whose derivatives vanish at
We denote the space
@M
C
of
C
via Fourier
is complete.
G a filter of all O-neighborhoods in
C’
and H
By [4, Ch. 2.12, Lena 3], there exists
{A+B; A F, B G}.
which is Cauchy in the topology inherited
m N such that H induces a filter H on
m
m
converges uniformly
On each ball {x
R
n}, r > 0, the filter
from
n,
@C"
pointwise to a function f
[4,
Oh. 2.9, Prop.
PROOF.
1]
@M
0 -q
and
[4,
Hence
space is ultrabornological.
@ is
m
on the subset
m
of
@C
and by
the filter F converges to f.
By Exercise 9 in
The space
Hm
Then f adheres to H
m
The spaces
THEOREM 4.
Ix[
@M are
Ch.
@M
ultrabornological.
3.15],
the strong dual of a complete Schwartz
is ultrabornological by Theorems
ultrabornological as an inductive limit of
Frchet
I, 2, and 3.
spaces
q N.
THEOREM 5.
The spaces
@C
and its strong dual
@C
are both complete, reflexive,
and ultrabornological spaces.
PROOF.
The space
@M
is complete as a strong dual of a bornological space.
the Fourier transformations
F:
OM C
and F:
M C
Since
are topological isomorphisms,
Theorem 5 follows from Theorems i, 3, and 4.
REFERENCES
Thorie des Distributions, Hermann, Paris, 1966.
i.
SCHWARTZ, L.
2.
GROTHENDIECK, A. Produits, Tensoriels Topologiques et Espaces Nuclaires, Memoirs
of the AMS 16 Providence (1955).
ULTRABORNOLOGICAL SPACES
816
Topological Vector Spaces, Grad. Texts in Math., (3), 3rd
printing, Springer 1971.
3.
SCHAEFER, H.
4.
HORVTH,
J.
opolo$ical Vector Spaces and Distributions, Addison-Wesley,
Reading 1966.
5.
KUCERA, J.
On Multipliers of Temperate Distributions, Czech. Math. J. 21, (96),
(1971), 610-618.
6.
KUCERA, J., MCKENNON, K.
Certain Topologies on the Space of Temperate
Distributions and its Multipliers, Indiana Univ. Math. J. 24 (8), (1975),
773-775.
7.
KUCERA, J., MCKENNON, K. The Topology on Certain Spaces of Multipliers of
Temperate Distributions, Rocky Mountain J. of Math. ! (2) (1977), 377-383.
8.
KUCERA, J., BOSCH, C. Bounded Sets in Fast Complete Inductive Limits,
Intern. J. Math. 7 (3) (1984), 615-617.