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A note on tensor products of Banach spaces
JOURNAL
OF MATHEMATICAL
ANALYSIS
AND
37, 235-238 (1972)
APPLICATIONS
A Note on Tensor Products of Banach Spaces NGUYEN Department
PHUONC
CAc
of Mathematics, The University Iowa City, Iowa 52240
of Iowa,
Submitted by Ky Fan
1. Let G and H be two Banach spaces. Let p be any real number with 1 < p < cc. A cross-norm r2, can be defined on G @ H as follows [l]: For each u = Cy=, xi @yi (xi E G, yi E H; i = l,..., n), put
where G’ is the dual of G, l/p + l/p* = 1 and the infimum is taken over all possible representations of u of the form u = Cy=, xi @ yi . G BD H denotes the completion of G @ H equipped with the norm rrP. For K= 1, 2, let (Mk, Zk,pcLk) b e measure spaces with positive Radon measures pK . Lp(Mk , plc) is the familiar Banach space of (equivalence classes) of functions on n/r having integrable p-th power. If F is a Banach space, then Lp[Mk , ps; F] denotes the corresponding space of F-valued functions. S. Chevet [l] has proved thatL”(M, , pl) @, L”(Ms , ps) could be identified
with -W4 , I+; WW , PJI. 2. each
In this note we define another cross-norm
r,, on G @ H by putting
for
where, again, the infimum is taken over all possible representations of u. It can be verified by the standard method that rO is a cross-norm (see e.g. [6, ExposC 21, where it is proved that nr is a cross-norm). G Qj, H denotes the completion of G @ H equipped with rO . The tensor product G @a H is related to the co-nuclear mappings as defined in [3].
235 0 1972 by Academic Press, Inc.
236
cAc
Let c0 be the well-known Banach Using Chevet’s method [l], it can for a Banach space F, c,[F] denotes converging to 0 and equipped with
space of all sequences converging to 0. be proved that c,, @jj, cc,g c,,[c,,], where, the space of all F-valued sequences (xi) the norm
II(X
We will show that, technique. Let A be a locally We denote by C,[A; which given any c >
=
SUP i
‘1 .
in fact, a more general statement holds, using a different compact Hausdorff space and let E be a Banach space. E] the space of all continuous, E-valued functions f for 0 there exists a compact subset K of A such that
IIfWll d 6 We provide
i/ Xi
x $ K.
C,[A; E] with the norm
If E is the complex field C, we write C,(A) for C,,[A; C]. We will prove the following.
PROPOSITION1. C,,(A) Bj, E can be identified with C,,[A; E]. 3. Before proving Prop. 1, we have to establish two results. A linear mapping T from a Banach space G into a Banach space His calledp-absolutely summing (or quasi p-integral) [3] if there exists a number p such that for any finite family {xi}Tzl of elements of G we have
(1) Let &o(T) be the smallest number p for which (1) holds. Provided with the norm &,Q, the space I,O(G, H) of all p-absolutely summing mappings from G into H is a Banach space [3].
LEMMA 1. Let G and H be two Banach spaces.The normed dual of G B,, H can be identified with the Banach space of all l-absolutely summing mappings ,from G into the dual H’ of H.
TENSOR PRODUCTS OF BANACH SPACES
237
Proof. We note that the dual of c,[H] is the Banach space P[H’] of all H’-valued sequences (ri’) such that
ll(Yi’>ll= f IlYi’ II < co* i=l
Otherwise the proof is just a repetition of a proof given in [5] of the fact that the dual of G gj, H is the space of all p*-absolutely summing mappings from G into H’. For the definitions and the properties of nuclear mappings and the crossnorms r and Eon the tensor product of two Banach spaces,we refer the reader to [2] or [6]. It can be proved [I] that the projective norm rr is equal to rr, . LEMMA
l-absolutely
2. A nuclear mapping T from C,,(A) into a Banach space F is summing and the nuclear norm v(T) is equal to h,o( T).
Proof. When the space A is compact, the statement is a consequence of Prop. 43 and 47 of [3]. With a number of easy modifications in the argument of [3], it can be proved that the Lemma also holds for locally compact spaces A. Proof of Proposition 1. We first prove that on C,,(A) @E the norm v,, coincides with the norm E, i.e., the norm induced by the Banach space of all continuous linear mappings from [C,(A)]’ into E. In fact, since ([4, p. 1331) the dual system (C,,(A) @ E, [C,(A)’ @ E’]) is separable, by Lemmas 1 and 2 and because of the fact that [C,(A)]’ BE’ is a subspace of the space [C,(A)]’ & E’ of a11nuclear mappings from C,,(A) into E’, on C,,(A) @ E the topology defined by v,, is the topology of uniform convergence on the unit ball of [C,(A)]’ &, E’. But it is well known that the dual of [C,,(A)]’ @j,, E’ is the Banach space of all continuous linear mappings from[C,,(A)]’ into E”, the bidual of E. Hence on C,,(A) @ E the crossnorms ~a and E coincide. Consequently C,,(A) &, E = C,(A) Bj, E. But C,(A) @, E = CJA; E] (see e.g. [2, Chap. 1, p. 901). The proof is complete. Proposition 1 is based on the fact that for every Banach space E,
We shall see that the spaceLi(M, p) behaves similarly to C,(A) in this respect. PROPOSITION 2. Let p be a positive Radon measure on the measurable space M. Then for every Banach space E we have
E 0, L1(M, p) = E 0, L1(M, CL).
C.&C
238
Proof. By Lemma 1, the dual of E $&Ll(M, p) is the space of all l-absolutely summing mappings from E into L”(M, CL). Since L”(M, CL) has the extension property [3], by Prop. 38 and 43 of [3], every nuclear mapping T from E into Lm(M, p) is l-absolutely summing and &o(T) is equal to the nuclear norm v(T). The proof can be completed as for Prop. 1. Note. We do not know whether the spaces Lr’ (or spaces “closely alike”) are the only Banach spaces E such that Lp(M, p) Bj, E = Ln[M, p; E]. If p = 1, then it is well-known that
L1(M, p) B E = L1(M, p) @ E = L1[M, /L; E] n
1
(cf., e.g., [6]) holds for every Banach space E. Added in Proof: After this paper has been submitted for publication, a paper ‘Produits tensoriels d’espaces de Banach’ by P. Saphar appeared in Studio Math. 38 (1970), 71-100. In that paper both of our Prop. 1 and 2 are proved with a different method. Also, for Prop. 1, C, is replaced by the space C of continuous functions on a compact space.
REFERENCES
1. S. CHEVET, Sur certains produits tensoriels topologiques d’espaces de Banach, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 11 (1969), 120-138. 2. A. GROTHENDIECK, Produits tensoriels topologiques et espaces nucleaires, Mem. Amer. Math. Sot. 16 (1955), page 28 and page 88. 3. A. PERSSON AND A. PIETSCH, p-nukleare und p-integrale Abbildungen in Banachrluman, Studiu Math. 33 (1969), 19-62. vector spaces, “Cambridge 4. A. P. ROBERTSON AND W. ROBERTSON, Topological Tracts No. 53,” Cambridge University Press, Mass., 1964. 5. P. SAPHAR, Produits tensoriels topologiques et classes d’applications lineaires, C.R. Acad. Sci. Paris Ser. A 266 (1968), 526-528. 6. L. SCHWARTZ, Produits tensoriels topologiques, Seminar Notes, Secretariat mathematique, Paris, 1953-1954.
OF MATHEMATICAL
ANALYSIS
AND
37, 235-238 (1972)
APPLICATIONS
A Note on Tensor Products of Banach Spaces NGUYEN Department
PHUONC
CAc
of Mathematics, The University Iowa City, Iowa 52240
of Iowa,
Submitted by Ky Fan
1. Let G and H be two Banach spaces. Let p be any real number with 1 < p < cc. A cross-norm r2, can be defined on G @ H as follows [l]: For each u = Cy=, xi @yi (xi E G, yi E H; i = l,..., n), put
where G’ is the dual of G, l/p + l/p* = 1 and the infimum is taken over all possible representations of u of the form u = Cy=, xi @ yi . G BD H denotes the completion of G @ H equipped with the norm rrP. For K= 1, 2, let (Mk, Zk,pcLk) b e measure spaces with positive Radon measures pK . Lp(Mk , plc) is the familiar Banach space of (equivalence classes) of functions on n/r having integrable p-th power. If F is a Banach space, then Lp[Mk , ps; F] denotes the corresponding space of F-valued functions. S. Chevet [l] has proved thatL”(M, , pl) @, L”(Ms , ps) could be identified
with -W4 , I+; WW , PJI. 2. each
In this note we define another cross-norm
r,, on G @ H by putting
for
where, again, the infimum is taken over all possible representations of u. It can be verified by the standard method that rO is a cross-norm (see e.g. [6, ExposC 21, where it is proved that nr is a cross-norm). G Qj, H denotes the completion of G @ H equipped with rO . The tensor product G @a H is related to the co-nuclear mappings as defined in [3].
235 0 1972 by Academic Press, Inc.
236
cAc
Let c0 be the well-known Banach Using Chevet’s method [l], it can for a Banach space F, c,[F] denotes converging to 0 and equipped with
space of all sequences converging to 0. be proved that c,, @jj, cc,g c,,[c,,], where, the space of all F-valued sequences (xi) the norm
II(X
We will show that, technique. Let A be a locally We denote by C,[A; which given any c >
=
SUP i
‘1 .
in fact, a more general statement holds, using a different compact Hausdorff space and let E be a Banach space. E] the space of all continuous, E-valued functions f for 0 there exists a compact subset K of A such that
IIfWll d 6 We provide
i/ Xi
x $ K.
C,[A; E] with the norm
If E is the complex field C, we write C,(A) for C,,[A; C]. We will prove the following.
PROPOSITION1. C,,(A) Bj, E can be identified with C,,[A; E]. 3. Before proving Prop. 1, we have to establish two results. A linear mapping T from a Banach space G into a Banach space His calledp-absolutely summing (or quasi p-integral) [3] if there exists a number p such that for any finite family {xi}Tzl of elements of G we have
(1) Let &o(T) be the smallest number p for which (1) holds. Provided with the norm &,Q, the space I,O(G, H) of all p-absolutely summing mappings from G into H is a Banach space [3].
LEMMA 1. Let G and H be two Banach spaces.The normed dual of G B,, H can be identified with the Banach space of all l-absolutely summing mappings ,from G into the dual H’ of H.
TENSOR PRODUCTS OF BANACH SPACES
237
Proof. We note that the dual of c,[H] is the Banach space P[H’] of all H’-valued sequences (ri’) such that
ll(Yi’>ll= f IlYi’ II < co* i=l
Otherwise the proof is just a repetition of a proof given in [5] of the fact that the dual of G gj, H is the space of all p*-absolutely summing mappings from G into H’. For the definitions and the properties of nuclear mappings and the crossnorms r and Eon the tensor product of two Banach spaces,we refer the reader to [2] or [6]. It can be proved [I] that the projective norm rr is equal to rr, . LEMMA
l-absolutely
2. A nuclear mapping T from C,,(A) into a Banach space F is summing and the nuclear norm v(T) is equal to h,o( T).
Proof. When the space A is compact, the statement is a consequence of Prop. 43 and 47 of [3]. With a number of easy modifications in the argument of [3], it can be proved that the Lemma also holds for locally compact spaces A. Proof of Proposition 1. We first prove that on C,,(A) @E the norm v,, coincides with the norm E, i.e., the norm induced by the Banach space of all continuous linear mappings from [C,(A)]’ into E. In fact, since ([4, p. 1331) the dual system (C,,(A) @ E, [C,(A)’ @ E’]) is separable, by Lemmas 1 and 2 and because of the fact that [C,(A)]’ BE’ is a subspace of the space [C,(A)]’ & E’ of a11nuclear mappings from C,,(A) into E’, on C,,(A) @ E the topology defined by v,, is the topology of uniform convergence on the unit ball of [C,(A)]’ &, E’. But it is well known that the dual of [C,,(A)]’ @j,, E’ is the Banach space of all continuous linear mappings from[C,,(A)]’ into E”, the bidual of E. Hence on C,,(A) @ E the crossnorms ~a and E coincide. Consequently C,,(A) &, E = C,(A) Bj, E. But C,(A) @, E = CJA; E] (see e.g. [2, Chap. 1, p. 901). The proof is complete. Proposition 1 is based on the fact that for every Banach space E,
We shall see that the spaceLi(M, p) behaves similarly to C,(A) in this respect. PROPOSITION 2. Let p be a positive Radon measure on the measurable space M. Then for every Banach space E we have
E 0, L1(M, p) = E 0, L1(M, CL).
C.&C
238
Proof. By Lemma 1, the dual of E $&Ll(M, p) is the space of all l-absolutely summing mappings from E into L”(M, CL). Since L”(M, CL) has the extension property [3], by Prop. 38 and 43 of [3], every nuclear mapping T from E into Lm(M, p) is l-absolutely summing and &o(T) is equal to the nuclear norm v(T). The proof can be completed as for Prop. 1. Note. We do not know whether the spaces Lr’ (or spaces “closely alike”) are the only Banach spaces E such that Lp(M, p) Bj, E = Ln[M, p; E]. If p = 1, then it is well-known that
L1(M, p) B E = L1(M, p) @ E = L1[M, /L; E] n
1
(cf., e.g., [6]) holds for every Banach space E. Added in Proof: After this paper has been submitted for publication, a paper ‘Produits tensoriels d’espaces de Banach’ by P. Saphar appeared in Studio Math. 38 (1970), 71-100. In that paper both of our Prop. 1 and 2 are proved with a different method. Also, for Prop. 1, C, is replaced by the space C of continuous functions on a compact space.
REFERENCES
1. S. CHEVET, Sur certains produits tensoriels topologiques d’espaces de Banach, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 11 (1969), 120-138. 2. A. GROTHENDIECK, Produits tensoriels topologiques et espaces nucleaires, Mem. Amer. Math. Sot. 16 (1955), page 28 and page 88. 3. A. PERSSON AND A. PIETSCH, p-nukleare und p-integrale Abbildungen in Banachrluman, Studiu Math. 33 (1969), 19-62. vector spaces, “Cambridge 4. A. P. ROBERTSON AND W. ROBERTSON, Topological Tracts No. 53,” Cambridge University Press, Mass., 1964. 5. P. SAPHAR, Produits tensoriels topologiques et classes d’applications lineaires, C.R. Acad. Sci. Paris Ser. A 266 (1968), 526-528. 6. L. SCHWARTZ, Produits tensoriels topologiques, Seminar Notes, Secretariat mathematique, Paris, 1953-1954.