Geometry of integral polynomials, M-ideals and unique norm preserving extensions
We use the Aron-Berner extension to prove that the set of extreme points of the unit ball of the space of integral k-homogeneous polynomials over a real Banach space X is {±φ k:φ∈X *, ||φ||=1}. With this description we show that, for real Banach spaces X and Y, if X is a nontrivial M-ideal in Y, the...
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| Autores principales: | , , |
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| Formato: | JOUR |
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00221236_v262_n5_p1987_Dimant |
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| Sumario: | We use the Aron-Berner extension to prove that the set of extreme points of the unit ball of the space of integral k-homogeneous polynomials over a real Banach space X is {±φ k:φ∈X *, ||φ||=1}. With this description we show that, for real Banach spaces X and Y, if X is a nontrivial M-ideal in Y, then ⊗̂ εk,s k,sX (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M-ideal in ⊗ ̂εk,s k,sY. This result marks up a difference with the behavior of nonsymmetric tensors since, when X is an M-ideal in Y, it is known that ⊗̂ εk kX (the k-th tensor product of X endowed with the injective tensor norm) is an M-ideal in ⊗̂ εk kY. Nevertheless, if X is also Asplund, we prove that every integral k-homogeneous polynomial in X has a unique extension to Y that preserves the integral norm. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given. © 2011 Elsevier Inc. |
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