445
I nternat.. J. Math. Math. S ci.
Vol. 3 No. 3 (1980) 445-454
OPERATOR REPRESENTATION OF WEAKLY CAUCHY SEQUENCES IN
PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES
J.M. BAKER
Department of Mathematics
Western Carolina University
Cullowhee, North Carolina 28723
(Received August 22, 1979)
ABSTRACT.
It is shown that the above sequences always determine linear trans-
formations and if the sequences are bounded under the least cross norm, that the
transformations are continuous.
Such operators are characterized to within
algebraic isomorphism with the weak-star sequential closure of the tensor product
space in its second dual, and consequently certain classes of weakly sequentially
complete projective tensor products are exhibited.
KEY WORDS AND PHES.
Tenor product, Weak topology, Opator, Sequential
im
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i
46B05, 47B05.
INTRODUCTION.
Let E and F be normed linear spaces and G and H subspaces of their duals E*
and F*, respectively.
Let % be the least cross norm [8] (operator norm, norm
giving the inductive topolgy), and consider E
$%F,
the %-completion of the tensor
J.M. BAKER
446
product E
F)* by
(E
xj
(R)xF.
m E
(El gi @ hi)(Z
F.
yj
Then G (R) H is algebraically isomorphic to a subspaee of
As such G
yj) -i,jZ gi(xj)hi(Yj)
xj
H carries the dual norm
where
H,
G’hi
gi
of % which is itself
a cross norm different, in general, from the greatest cross norm
y[8] (nuclear
norm, norm giving the projective topology).
It is easy to show that in order for E
F to be
o(E
F, G
H)-
%0
sequentially complete (or sequentially complete in its weak topology) it is
necessary that E and F be (weakly) sequentially complete in their respective
weak topologies o(E,G) and o(F,H).
l’
To motivate our work, let E
the
space of absolutely summable sequences, G--c, the space of convergent
sequences embedded in the bounded sequences, and F a weakly sequentially
complete Banach space.
o(
F, c
0
Then
i% F
is sequentially complete in its
F*) topology, and the canonical
map @% F
/
e(c,F) is
surective, where L(c,F) is the space of bounded linear transformations from
c into F.
This example and others of -tensor products which are sequentially
complete under a weak topology and coincide with an associated operator space
are found in [i].
Similar examples exist where the sequential completeness is
under the weak topology.
For instance, let E and F be reflexive Banach spaces
with bases such that every operator from E* into F is compact.
reflexive
[4, p. 188], whence E
%F
Then E
is weakly sequentially complete.
since F has a basis it also has the approxlmatlor[ property (a.p.)
so that the canonical
inection E
F
/
L(E,F)
%
F is
Moreover,
[7, p. 115]
C(E*,F) is surJ ective,
where C(E*,F) is the space of compact operators from E* into F.
Thus, we will show that the above examples are special cases of more
general properties enjoyed by weakly Cauchy sequences in projective tensor
product spaces and that the equivalence class of such a sequence (definition
follows)alays defines a linear transformation from G into H* (Theorem i)
447
WEAKLY CAUCHY SEQUENCES IN BANACH SPACES
which is continuous if the equivalence class is, in a sense, bounded (Theorem
3).
Further, if G and H contain the extreme points of the unit cells
SF,,
we show that these equivalence classes are, algebraically, precisely the
functionals in the weak star sequential closure of E
@
SE,
and
F in its second dual
and if H is norm-closed that these functlonals are operators from G into
Consequently, for reflexive spaces E and weakly sequentiallycomplete
(Theorem 5).
spaces F such that every operator from
@
E
E*
into F is compact, we obtain that
F is weakly sequentially complete if E or F has the a.p. (Corollary 6).
In addition to notation already introduced, J will be the usual embedding
of a normed linear space F in its second dual
F
star
/
H*
is defined by
(O(F**, F*))
complete.
Thus,
h(y), y
F, h e H.
sequential closure of JF in
sequential closure of
our work).
(y)h
F
,(F)
in
H*
If H is a subspace of
F**,
and
F*
K(F) will be the weak-
(F)
the
(this last space arising in a natural way in
K(F) and
(F)
F if F is O(F,H)-sequentlally
The sense of the last equality (algebraic isomorphism, homeomorphism,
isometric isomorphism) depends on results in [I0] and its bibliography which
can be used to generate corollaries to Theorems 1,3 and 5.
RESULTS:
It is simple to show that G and H are total subspaces (G
{0}) of the
O
duals of the normed linear spaces E and F, respectively, if and only if o(E @ F,
G @ H) is Hausdorff on E @ F.
E
(R)
Also, if at least G is total over E, then
F is algebraically isomorphic to a subspace of the linear transformations
from G into F by
(Y. x i @
where x
i
E
E,
Yi
E
Yi)g =g(xi)Yi
F, g e G.
We define (E,F,G,H) to be the set of equivalence classes of
o(E
F, G
H)
Cauchy sequences in E @ F where equivalence of sequences
448
J.M. BAKER
means agreement in the limit at points of G @ H.
This becomes a vector space
when given the natural addition and scalar multiplication and as such contains
a copy of E 0 F:
E
e F
t
/
-+
(E,F,G,H)
defined by
THEOREM I:
of
E* and H
z where (t,t,t,
z
Let E and F be normed linear spaces, G a total subspace
a norm-closed total subspace of
F*.
Then (E,F,G,H) is
algebraically isomorphic to a subspace of the linear transformations from
Let {z i}
PROOF:
(E,F,G,H), where
z
s
zi=r.
i
Yk, i
k=l xk’
for
Xk, i
E,
Yk,i
F, i
1,2,-.-
Then if
> 0, g
G and h
H there
exists N > 0 such that
h(
s
n
k=l
s
m
gCxk n ) Yk, n
for m,n > N.
Fixing g
n
the sequence {X g(x
k
i
g
k=l
G and varying
l)Yk
k-I
h(
(Xk,m)Yk, roll
< e
and h, we see that for each g
} in F is (F,H)
G
Cauchy.
Thus
h*--cr(H*,H)
i
limi ,IF.k=lg(xk,l)yk,i]
is uniquely defined since H is total, is independent of the choice {z
and lles in
(F).
Define (U(z))g
h*.
z,
It is straight forward that U(z) is
a linear (not necessarily continuous) map from G into
linear.
i}
(F)
and that U is
WEAKLY CAUCHY SEQUENCES IN BANACH SPACES
Put U(z)
(E,F,G,H).
0, z
449
G, and consequently for
Then for all g
H,
every h
i
r. g(x
k
k=l
llm
i
si
i_
{k=ZlXk,
showing that
G (R) H) topology.
(R)
Yk, i }
i)h(Yk, i)- 0
converges to the null sequence in the o(E @ F,
0 and U is inJective.
Thus, z
E*
The following points out that if G
F*,
and H
every functional in
K(F) can be reached by a Um map.
PROPOSITION 2:
Let E and F be normed linear spaces.
property that given
x*
I x*II
E*
sequence {z i}
i
m(E,F,E*,F*)
K(F) there exists z
I, such that
(z))x*
y**.
z such that for all cross norms
and
Further, there exists a
on E @ F,
T
T(zi)
"II Y**II
1,2,
PROOF:
I xll
{yn} c__ F
Let
I, and let z i
E @ F.
There exists
y**.
(U(z))x*
I[
converge
x @
x*
e
Yi"
o(F**,F*)
Then {z
i}
SE, I x*ll
to
y**
is o(E @
F,
i, such that
K(F), fix x
E* @ F*)
x*(x) i,
E,
Cauchy in
whence
[5, Lemma 2] holds for normed linear spaces, we may
Since
IIY**II
assume
norm
y**
Then U has the
Ynll
so that T(z
n) =11 xll
llYnll
IIY**II
for any cross
T.
Below,
that sup
i
A(E,F,G,H)
A(z i)
TEOR4 3:
<
+
will denote the subspace of those z
for some {z
i}
(E,F,G,H) such
e z.
Let G and H be total over the normed linear spaces E and F
and H be norm closed in
F*.
Then
A(E,F,G,H)
L(G,(F)) by the mapping
I (z) ll !{z nf z sup
(z).
i
i}
a subspace of
E
U, and
is algebraically isomorphic to
J.M. BAKER
450
Let {z
s
PROOF:
i}
z e
ml(E,F,G,H),
i
_ E Xk,i @ Yk,i’
k 1
zi
with
i-- 1 2,
Then
s
[I (u(z))ll
sup
geSG
i
-sup
geSG
sup
hS
H
i
k-I
_<
sup
llm
l(zl)
sup
geS G heS H
<_
i
sup A (z i)
i
The central result, Theorem 5, provides an algebraic characterization
of the weak-star sequential closure of E
F in (E
@
@l
F)**
(i.e. in the
dual of the space of integral bilinear forms on E x F [3]).
on [6] and [9]
LEMMA A:
and we cite [6] explicitly:
(Rainwater):
Let {x } be a norm bounded sequence in a normed
linear space X and M the set of extreme points of
then {x
LEMMA 4:
If {x } is M-Cauchy,
Let X be a normed linear space and W a total subspace of
Sx,,
which contains the extreme points of
n}
Then {Jx
and
PROOF:
are
o(X**,X*)
THEOREM 5:
subspaces of
E*
SF,
{Wk}
where
W2k_l
X*
norm bounded
for
eery
f e W.
convergent to the same functional in
Both sequences converge in
Cauchy sequence
and
{Jyn }
{Xn }’ {yn } two
f(xn)
llm f(yn
and
(X,W)-Cauchy sequences in X such that llm
SE,
SX,.
is weakly Cauchy.
n
of
The proof keys
X**
xk, W2k
X**.
to the same limit as the weakly
Yk’
k
1,2,
.-..
Let E and F be normed linear spaces with G and H total
and
F*,
then
respectively
I(E,F,G,H)
If also H is norm closed in
F*,
If G and H contain the extreme points
is algebraically isomorphic to
K(E
8
K(E
@.
F).
F) is algebraically isomorphic to a
451
WEAKLY CAUCHY SEQUENCES IN BANACH SPACES
subspace of
L(G,(F))
withll TII <_ I.
by a mapping T which is continuous
If
G and H determine the norm in E and F (i.e. have Dixmier characteristic one),
The extreme points of the unit ball of (E
PROOF:
those functionals of the form
SE,
SF,,
and
respectively [9].
that sup %(z
i)
m%(E,F,G,H)
K(E
0%
F) by V(z)
8%
and
J(zi)
and choose {z
m%(E,F,G,H)
Thus, for some z, {z i} e z e
F.
8
F) onto
@
into
mk (E ,F ,G ,H)
L(G,(F)).
UV
For if
z**E
(E,F,G,),
-I
-I.
z, where
{J-!
L(G,(F))).
(t), j-I (t), --.}z.
Recalling that the action of U(z) on g
sequence
J(
z, we choose {J
{z i}
@
K(E
8
F)
Let {z i}
[[z**[[
ence
E
E
@%
T(J(E
is an algebraic
Thus, T is the required
r
-Iz** z**
y
I
yk ))
E
(R)
F,
@%
F) is J
J(E
Let t
Consider U(z)
(t), --.}z.
[g()yk
@
Yk for
F converge weak-star to
[5] and from
in E and F and x
@
z**.
and V(z)
@
-I
(consider-
F).
Now
L(G,(F)).
G is independent of the choice of
(t),J
o(H*,H)
yk ), (U(Z))g
G,
for each g
Tz**
Define
and by Theorem 3 U is inJective
We take T
F algebraically embedded in
V -I (t)
%(z i)
F.
O%
By Lemma 4, V is well-defined, and clearly
isomorphism, and we claim that T when restricted to J(E
t
e z so
is the weak-star limit of a weakly Cauchy, hence norm bounded,
isomorphism of K(E
@%
i}
where the limit is in the
To establish the second claim, consider that V
ing E
are precisely
are extreme points of
inJective. Moreover, V is surjective.
sequence {z i} in E
from
y*
is weakly Cauchy in E
llm
F)**.
x*
(E,F,G,H)
Let z
/
it is linear and
z**
where
+ =. By Lemma A, {z i}
weak-star topology of (E
then
y*,
8
<
i
V:
x*
F)*
0%
z**
Then if
( @ yk)g
every
in K(E
@%
F).
@
Yk
E
E
@%
F.
We y take
3 obtain
whence
]T! i.
If G
a
H deteine the norms
452
J.M. BAKER
gS
geS
G
G
sup
sup
g(x) h(y)
gS G hcS
H
sup
sup
g(x) h(y)
(x s y)
Lt E and F be Banach spaces such that every
COROLLARY 6:
from
E*
into F is compact with E or F having the a.p.
and F is weakly sequentially complete, E
PROOF:
The a.p. insures that E
compact operators from
E*
Theorem 5 use G
E*
into F
and H
,
F*.
operator
Then if E is reflexive
F is weakly sequentially complete.
F coincides with the space of
p. 113], whence
L(E*,F)
E
Since the weak-star sequential closure of
a normed linear space is equal to that of its norm completion, K(E
is algebraically isomorphic under T to a subspace of
restricted to J(E
(R) F),
is the canonical map
D
F.
J(E
Since T, when
F into
F of K(E
F)
L(E*,F),
(R)
it
F), is
Thus,
F)
__c TK(E F) !L(E*,F)
where each inclusion is isometric.
isomorphism of K(E
L(E*,F).
of E
follows that T, when restricted to the subspace E
extended to E
In
F.
F) onto E
E
F
It follows that T is an isometric
F, completing the proof.
Theorem 5 and Corollary 6 give information in a variety of special cases.
For instance, every operator from a reflexive space E into
[2, p. 515], and
I, by having a Schauder basis,
I
has the a.p.
is compact
Thus,
WEAKLY CAUCHY SEQUENCES IN BANACH SPACES
EXI
is weakly sequentially complete by Corollary 6.
the spaces
q
q
>r > I
q-1
into
r
In particular,
are weakly sequentially complete (q > i)
X
the spaces
q
weakly sequentially complete.
qXr
453
X
r
For
are reflexive [4, p. 189], and thus
However, if p
q
q
_< r, the spaces
i
are not reflexive and there exists a non-compact operator from
[4, p. 189].
Thus
L(p,4 + q X r"
r/r_i
Therefore,
"-q
r
L
p
But by [3, p. 122] we have
so
is not weakly sequentially complete,
q
q-i
< r.
Following Theorem 5 one naturally seeks conditions under which T is
Among the more interesting questions Is that of K(E 0 F)
X
being homeomorphic to the whole of L(G,(F)). Aside from Corollary 6, in
a homeomorphlsm.
dealing with some of the possibilities surrounding G and H we obtain results
which are accessible through the piecing together of several theorems in
For instance, let E be a Banach space with separable dual E
Put G
reflexive Banach space.
can show via Theorem 5 or
E*
and It
[3] that K(E
@X
F*.
If
E*
or
F*
[3].
and F a separable
has the a.p. one
F) is linearly homeomorphic to L(E*,F).
The author wishes to acknowledge his major professor, Dr. R.D. McWilliams,
for his continuing encouragement and support since graduation from Florida
State University, 1969.
REFERENCES
i.
J.M. Baker,
We
sequential completeness in spaces of operators,
Proc. A.M.S. 25 (1970), 193-198.
2.
N. Dunford and J.T. Schwartz, Linear Op.erator.s, Part I, Intersclence,
N.Y., 1958.
3.
A. Grothendieck, Produits tensoriels to/logiques e_t spaces nucleaires,
Memoirs A.M.S. 16
(1966).
J.M. BAKER
454
4.
J.R. Holub, Hilbertian operators an__d reflexive tensor products, Pacific
J. Math. 36 (1971), pp. 185-194.
5.
R.D. McWilliams, A note on weak sequential convergence, Pacific J.
Math 12 (1962), 333-335.
6.
J. Rainwater, Weak convergence of bounded sequences
Proc. A.M.S. 14
(1963), 999.
7.
H.H. Schaefer, Topologica.l Vector Spaces, 2nd ed., MacMillan, N.Y., 1967.
8.
R. Schatten, A
9.
I. Singer, Le___s points extremaux de la boule
tensoriel
(1958)
i0.
I. Singer
theorv of cross spaces,
Annals of Math. Studies 26 (1950).
unit
norm inductif d’espaces de Banach,
du dual d’un produit
Bull. Sci. Math. (2) 82
73-80.
Weak compactness, pseudo-reflexivity, and
asi-
reflexivity, Math. Ann. 154 (1964), 77-87.
Department of Mathematics, Western Carolina University, Cullowhee, NC
28723