OPERATOR REPRESENTATION OF WEAKLY CAUCHY SEQUENCES IN PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES

It is shown that the above sequences always determine linear transformations 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.

product E (R)xF.Then G (R) H is algebraically isomorphic to a subspaee of (E F)* by (El gi @ hi)(Z xj yj) -i,jZ gi(xj)hi(Yj) where gi G'hi H, xj m E yj F. As such G H carries the dual norm of % which is itself a cross norm different, in general, from the greatest cross norm y[8] (nuclear norm, norm giving the projective topology).
show that in order for E F to be o(E F, G %0 H)- It is easy to 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).To motivate our work, let E l' the space of absolutely summable sequences, G--c, the space of convergent sequences embedded in the bounded sequences, and F a weakly sequentially Banach space.Then i% F is sequentially complete in its complete o( F, c 0 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.Then E % F is [4, p. 188], whence E % F is weakly sequentially complete.Moreover, reflexive since F has a basis it also has the approxlmatlor[ property (a.p.) [7, p. 115]   that the canonical inection 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) 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 SE, and SF,, we show that these equivalence classes are, algebraically, precisely the functionals in the weak star sequential closure of E @ F in its second dual and if H is norm-closed that these functlonals are operators from G into (Theorem 5).Consequently, for reflexive spaces E and weakly sequentiallycomplete spaces F such that every operator from E* into F is compact, we obtain that E @ 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 If H is a subspace of F* F / H* is defined by (y)h h(y), y F, h e H. K(F) will be the weak- star (O(F**, F*)) sequential closure of JF in F**, and (F) the sequential closure of F in H* (this last space arising in a natural way in our work).Thus, ,(F) K(F) and (F) F if F is O(F,H)-sequentlally complete.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 O {0}) of the duals of the normed linear spaces E and F, respectively, if and only if o(E @ F, G @ H) is Hausdorff on E @ F. Also, if at least G is total over E, then E (R) F is algebraically isomorphic to a subspace of the linear transformations from G into F by 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 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 -+ (E,F,G,H) defined by t / z where (t,t,t, z THEOREM I: Let E and F be normed linear spaces, G a total subspace of E* and H a norm-closed total subspace of F*.Then (E,F,G,H) is algebraically isomorphic to a subspace of the linear transformations from PROOF: Let {z i} z (E,F,G,H), where s zi=r.k=l xk' i Yk, i for Xk, i E, Yk,i F, i 1,2,-.-exists N > 0 such that Then if > 0, g

G and varying
and h, we see that for each g G n the sequence {X g(x k l)Yk i } in F is (F,H) Cauchy.

k-I
Thus i h*--cr(H*,H) limi ,IF.k=lg(xk,l)yk,i] is uniquely defined since H is total, is independent of the choice {z i} z, and lles in (F).Define (U(z))g h*.It is straight forward that U(z) is a linear (not necessarily continuous) map from G into (F) and that U is linear.
Put U(z) 0, z (E,F,G,H).Then for all g G, and consequently for every h showing that {k=ZlXk, i_ (R) Yk, i } converges to the null sequence in the o(E @ F, G (R) H) topology.Thus, z 0 and U is inJective.
The following points out that if G E* and H F*, every functional in K(F) can be reached by a Um map.
PROPOSITION 2: Let E and F be normed linear spaces.Then U has the property that given y** K(F) there exists z m(E,F,E*,F*) and * E* x I I x*II I, such that (z))x* y**.Further, there exists a sequence {z i} z such that for all cross norms T on E @ F, T(zi) "II Y**II i 1,2, PROOF: Let {yn} c__ F converge o(F**,F*) to y** K(F), fix x E, I I xll I, and let z i x @ Yi" Then {z i} is o(E @ F, E* @ F*) Cauchy in E @ F. There exists x* e SE, I I x*ll i, such that x*(x) i, whence (U(z))x* y**.Since [5, Lemma 2] holds for normed linear spaces, we may assume IIY**II I[ Ynll so that T(z n) =11 xll llYnll IIY**II for any cross norm T.
Below, A(E,F,G,H) will denote the subspace of those z E (E,F,G,H) such that sup A(z i) < + for some {z i} e z. i TEOR4 3: 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) is algebraically isomorphic to a subspace of L(G,(F)) by the mapping U, and I I (z) ll !nf sup (z).{z i} z i PROOF: Let {z i} z e ml(E,F,G,H), with 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]).The proof keys on [6] and [9] and we cite [6] explicitly: LEMMA A: (Rainwater): Let {x } be a norm bounded sequence in a normed linear space X and M the set of extreme points of SX,.If {x } is M-Cauchy, then {x is weakly Cauchy.n LEMMA 4: Let X be a normed linear space and W a total subspace of X* which contains the extreme points of Sx,, and {Xn }' {yn } two norm bounded (X,W)-Cauchy sequences in X such that llm f(x n) llm f(yn for eery f e W. Then {Jx n} and {Jyn } are o(X**,X*) convergent to the same functional in X**.
PROOF: Both sequences converge in X** to the same limit as the weakly Cauchy sequence {Wk} where W2k_l xk, W2k Yk' k 1,2, .-.. THEOREM 5: Let E and F be normed linear spaces with G and H total subspaces of E* and F*, respectively If G and H contain the extreme points of SE, and SF, then I(E,F,G,H) is algebraically isomorphic to K(E @.F).If also H is norm closed in F*, K(E 8 F) is algebraically isomorphic to a subspace of L(G,(F)) by a mapping T which is continuous withll TII <_ I.If G and H determine the norm in E and F (i.e. have Dixmier characteristic one), PROOF: The extreme points of the unit ball of (E 0% F)* are precisely those functionals of the form x* 8 y*, where x* and y* are extreme points of SE, and SF,, respectively [9].Let z (E,F,G,H) and choose {z i} e z so that sup %(z i) < + =.By Lemma A, {z i} is weakly Cauchy in E O% F. Define i V: m%(E,F,G,H) / K(E 0% F) by V(z) llm J(zi) where the limit is in the weak-star topology of (E 8% F)**.By Lemma 4, V is well-defined, and clearly it is linear and inJective.Moreover, V is surjective.For if z**E K(E 8 F) then z** is the weak-star limit of a weakly Cauchy, hence norm bounded, sequence {z i} in E 8 F. Thus, for some z, {z i} e z e (E,F,G,), and V(z) z**.
-I To establish the second claim, consider that V is an algebraic isomorphism of K(E @ F) onto mk (E ,F ,G ,H) and by Theorem 3 U is inJective from m%(E,F,G,H) into L(G,(F)).We take T UV-I.Thus, T is the required -I isomorphism, and we claim that T when restricted to J(E @% F) is J (consider- ing E @% F algebraically embedded in L(G,(F))).Let t J(E @ F).Now V -I (t) z, where {J-! (t), j-I (t), --.}z.Consider U(z) L(G,(F)).
Recalling that the action of U(z) on g G is independent of the choice of sequence {z i} z, we choose {J (t),J (t), --.}z.Then if t J( @ yk ), (U(Z))g o(H*,H) I [g()yk ( @ yk)g for each g G, ence T(J(E @ yk )) @ Yk for every @ Yk E E @% F. Let {z i} E E @% F converge weak-star to z** in K(E @% F).We y take %(z i) [[z**[[ [5] and from r 3 obtain Tz** -Iz** z** whence ]T! i.If G a H deteine the norms in E and F and x y (x s y) COROLLARY 6: Lt E and F be Banach spaces such that every operator from E* into F is compact with E or F having the a.p.Then if E is reflexive and F is weakly sequentially complete, E F is weakly sequentially complete.
PROOF: The a.p. insures that E F coincides with the space of compact operators from E* into F , p. 113], whence L(E*,F) E F. In Theorem 5 use G E* and H F*. Since the weak-star sequential closure of a normed linear space is equal to that of its norm completion, K(E F) is algebraically isomorphic under T to a subspace of L(E*,F).Since T, when restricted to J(E (R) F), is the canonical map of E F into L(E*,F), it follows that T, when restricted to the subspace E where each inclusion is isometric.It follows that T is an isometric isomorphism of K(E F) onto E 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 I is compact [2, p. 515], and I, by having a Schauder basis, has the a.p.Thus, EXI is weakly sequentially complete by Corollary 6.In particular, the spaces are weakly sequentially complete (q > i) For q X q >r > I the spaces are reflexive [4, p. 189], and thus q-1 q X r weakly sequentially complete.However, if p q q i _< r, the spaces q X r are not reflexive and there exists a non-compact operator from p into r [4, p. 189].Thus L(p,4 + q X r" But by [3, p. 122] we have r/r_i L so is not weakly sequentially complete, q Therefore, "-q r < r. q-i Following Theorem 5 one naturally seeks conditions under which T is a homeomorphlsm.Among the more interesting questions Is that of K(E 0 X F) being homeomorphic to the whole of L(G,(F)).Aside from Corollary 6, in   dealing with some of the possibilities surrounding G and H we obtain results which are accessible through the piecing together of several theorems in [3].
For instance, let E be a Banach space with separable dual E and F a separable reflexive Banach space.Put G E* and It F*.If E* or F* has the a.p. one can show via Theorem 5 or [3] that K(E @X 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.
m)Yk, k=l for m,n > N. Fixing g