Geometry of Spaces of Polynomials

“…The constants c n appearing in the following proof are the constants referred to in the Introduction (see [4], [9]). Also, we use the fact that k-linear forms are k-homogeneous polynomials.…”

Section: (Where If a Denominator Is Zero We Agree That The Fraction mentioning

confidence: 99%

A multilinear Phelps' Lemma

Aron

Cardwell

Garcı́a

et al. 2007

Proc. Amer. Math. Soc.

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“…Furthermore, this inequality is shown to be asymptotically sharp as n → ∞. In infinite dimensions inequalities for the norms of products of linear functionals have been presented in [1], [10] and [21].…”

Section: Introductionmentioning

confidence: 99%

The Norm of the Product of Polynomials in Infinite Dimensions

Boyd¹,

Ryan²

2006

Proceedings of the Edinburgh Mathematical Society

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Given a Banach space E and positive integers k and l we investigate the smallest constant C that satisfies P Q C P Q for all k-homogeneous polynomials P and l-homogeneous polynomials Q on E. Our estimates are obtained using multilinear maps, the principle of local reflexivity and ideas from the geometry of Banach spaces (type and uniform convexity). We also examine the analogous problem for general polynomials on Banach spaces.

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“…However, the weak * -closure of this set can be quite large, although { ϕ ⊗ · · · ⊗ ϕ : ||ϕ|| = 1} is norm-closed or even weakly-closed in particular cases [20]. For instance, if X is a Hilbert space then its w * -closure is the whole of B⊗ n,s,π s X * .…”

mentioning

confidence: 99%

An integral duality formula

Grecu¹,

Ryan²

2006

MATH. SCAND.

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We establish an integral formula for the duality between multilinear forms/homogeneous polynomials and tensor products for dual spaces with the approximation property and for which the injective tensor products of their preduals is separable and does not contain a copy of 1 . We deduce some multilinear Bishop-Phelps-type results.Although spaces of multilinear forms and homogeneous polynomials are dual spaces, multilinear analogues of the linear Bishop-Phelps Theorem [6] do not hold in general [2,14]. However, positive results exist. See [5] for a complete account and a detailed description of the problem. For an n-linear form defined on a product of Banach spaces X 1 × · · · × X n , two (in general different) types of norm attainment can be considered: as a linear functional defined on the projective tensor product X 1⊗π · · ·⊗ π X n or as an n-linear form on X 1 × · · · × X n . Under certain conditions, the duality between multilinear forms and tensor products can be expressed by means of an integral formula which will establish the equivalence of the two types of norm attainment and thus yield some multilinear Bishop-Phelps-type results. We note that the same formula, for Hilbert spaces, has already been applied to the study of other geometrical properties of spaces of tensors [11].1. Notation and terminology (see [10] and [19] for details). Given n (real or complex) Banach spaces X 1 , . . . , X n , we denote by X 1 ⊗ · · · ⊗ X n their tensor product and by π and ε the projective and injective norms respectively. If X 1⊗π · · ·⊗ π X n is the completion of X 1 ⊗ · · · ⊗ X n under the projective norm, then we have (. . , X n ), the space of continuous nlinear forms on X 1 × · · · × X n endowed with the supremum norm. We denote by. . , X n ) the spaces of integral and nuclear n-linear forms on X 1 × · · · × X n respectively. Let ⊗ n,s X be the n-fold symmetric tensor product of X. If we endow it with the topology inherited from X⊗ π · · · ⊗ π X and denote its completion by (⊗ n,s,π , the space of continuous symmetric n-linear forms on X × · · · × X endowed with the supremum norm.We denote by P ( n X) the space of all continuous n-homogeneous polynomials on X endowed with the natural supremum norm. The space ⊗ n,s,π

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Geometry of Spaces of Polynomials

Cited by 55 publications

References 9 publications

A multilinear Phelps' Lemma

A multilinear Phelps' Lemma

The Norm of the Product of Polynomials in Infinite Dimensions

An integral duality formula

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