ISSN:
1432-2064
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary In this paper we introduce new ⊗-norms on the tensor product of two Banach spaces E and F (in the sense of Grothendieck) which are given by $$P_k (u) = {\text{inf}}\left\{ {N{}_k[(yi)i] \cdot {\text{ }}\mathop {{\text{sup }}N_k }\limits_{\mathop {x'\varepsilon E'}\limits_{x' \leqq 1} } [(\left\langle {x_i ,\left. {x'} \right\rangle } \right.)i]} \right\}$$ where the infimum is taken over all representations $$\sum\limits_i {x{}_i \otimes y_i } $$ of u (finite sum; x i εE, y i ε F), N k denotes the usual norm on l k, k′ is the conjugate number of k and k ε[1,+∞]. The space E⊗ F with the norm P k will be denoted by E⊗F. These norms have also been introduced by P. Saphar. Moreover we give a representation of the elements of the completion of E⊗k F. We also prove, that the canonical injection of L k(X 1, ∇1, Μ 1) ⊗L k(X2, ∇2, Μ 2) into L k(X 1×X 2, ∇1 ⊗ℬ2, Μ1 ⊗ Μ2) can be extended as an isometry of $$L^k (X_1 ,\mathfrak{B}_1 ,\mu _1 )\hat \otimes _k L^k (X_2 ,\mathfrak{B}_2 ,\mu _2 ){\text{ onto }}L^k (X_1 \times X_2 ,\mathfrak{B}_1 \otimes \mathfrak{B}_2 (X_2 ,\mu _1 \otimes \mu _2 ).$$ onto Lk(X 1×X2, ∇, Μ2). Finally we introduoe “right and left k-nuclear operators”.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00531813
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