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1998 | 129 | 3 | 285-295
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The symmetric tensor product of a direct sum of locally convex spaces

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An explicit representation of the n-fold symmetric tensor product (equipped with a natural topology τ such as the projective, injective or inductive one) of the finite direct sum of locally convex spaces is presented. The formula for $⨂^n_{τ,s} (F_1⨁ F_2)$ gives a direct proof of a recent result of Díaz and Dineen (and generalizes it to other topologies τ) that the n-fold projective symmetric and the n-fold projective "full" tensor product of a locally convex space E are isomorphic if E is isomorphic to its square $E^2$.
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Bibliografia
  • [1] R. Alencar and K. Floret, Weak-strong continuity of multilinear mappings and the Pełczyński-Pitt theorem, J. Math. Anal. Appl. 206 (1997), 532-546.
  • [2] A. Arias and J. Farmer, On the structure of tensor products of $ℓ_p$-spaces, Pacific J. Math. 175 (1996), 13-37.
  • [3] F. Blasco, Complementación, casinormabilidad y tonelación en espacios de polinomios, doct. thesis, Univ. Compl. Madrid, 1996.
  • [4] F. Blasco, Complementation in spaces of symmetric tensor products and polynomials, Studia Math. 123 (1997) 165-173.
  • [5] J. Bonet and A. Peris, On the injective tensor product of quasinormable spaces, Results in Math. 20 (1991), 431-443.
  • [6] A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Math. Stud. 176, North-Holland, 1993.
  • [7] A. Defant and M. Maestre, Property (BB) and holomorphic functions on Fréchet-Montel spaces, Math. Proc. Cambridge Philos. Soc. 115 (1993), 305-313.
  • [8] J. C. Díaz and S. Dineen, Polynomials on stable spaces, Ark. Mat., to appear.
  • [9] S. Dineen, Complex Analysis on Infinite Dimensional Spaces, in preparation.
  • [10] K. Floret, Some aspects of the theory of locally convex inductive limits, in: Functional Analysis: Surveys and Recent Results II, K. D. Bierstedt and B. Fuchssteiner (eds.), North-Holland, 1980, 205-237.
  • [11] K. Floret, Tensor topologies and equicontinuity, Note Mat. 5 (1985), 37-49.
  • [12] W. T. Gowers, A solution to the Schroeder-Bernstein problem for Banach spaces, Bull. London Math. Soc. 28 (1996), 297-304.
  • [13] W. Greub, Multilinear Algebra, Universitext, Springer, 1978.
  • [14] A. Grothendieck, Produits tensoriels et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955).
  • [15] H. Jarchow, Locally Convex Spaces, Teubner, 1981.
  • [16] R. Ryan, Application of topological tensor products to infinite dimensional holomorphy, doct. thesis, Trinity Coll. Dublin, 1980.
  • [17] L. Schwartz, Théorie des distributions à valeurs vectorielles. I et II, Ann. Inst. Fourier (Grenoble) 7 (1957), 1-141, and 8 (1958), 1-209.
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bwmeta1.element.bwnjournal-article-smv129i3p285bwm
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