Denseness of Norm-Attaining Mappings on Banach Spaces

Choi, Yun Sung; Lee, Han Ju; Song, Hyun Gwi

doi:10.2977/prims/4

Cited by 7 publications

(5 citation statements)

References 26 publications

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“…In [AGM03] Aron, García and Maestre showed a polynomial Lindenstrauss theorem for the case of scalar-valued 2-homogeneous polynomials. This was extended to vector-valued 2homogeneous polynomials by Choi, Lee and Song [CLS10]. The aim of this work is to show a polynomial Lindenstrauss theorem for arbitrary degrees.…”

Section: Introductionmentioning

confidence: 97%

On the polynomial Lindenstrauss theorem

Carando

Lassalle

Mazzitelli

2012

Journal of Functional Analysis

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Under certain hypotheses on the Banach space $X$, we show that the set of $N$-homogeneous polynomials from $X$ to any dual space, whose Aron-Berner extensions are norm attaining, is dense in the space of all continuous $N$-homogeneous polynomials. To this end we prove an integral formula for the duality between tensor products and polynomials. We also exhibit examples of Lorentz sequence spaces for which there is no polynomial Bishop-Phelps theorem, but our results apply. Finally we address quantitative versions, in the sense of Bollob\'as, of these results

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Section: Introductionmentioning

confidence: 97%

On the polynomial Lindenstrauss theorem

Carando

Lassalle

Mazzitelli

2012

Journal of Functional Analysis

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“…For 2-homogeneous scalar-valued polynomials, the Lindenstrauss theorem was proved with full generality by Aron, García and Maestre in [8], where the Aron-Berner extension takes the place of the bitranspose. This result was later extended by Choi, Lee and Song [13] for vector-valued 2-homogeneous polynomials. In [11] a partial result was obtained for homogeneous polynomials of any degree.…”

Section: Introductionmentioning

confidence: 70%

Bounded Holomorphic Functions Attaining their Norms in the Bidual

Carando¹,

Mazzitelli²

2015

Publ. Res. Inst. Math. Sci.

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Under certain hypotheses on the Banach space X, we prove that the set of analytic functions in A u (X) (the algebra of all holomorphic and uniformly continuous functions in the ball of X) whose Aron-Berner extensions attain their norms, is dense in A u (X). This Lindenstrauss type result holds also for functions with values in a dual space or in a Banach space with the so-called property (β). We show that the Bishop-Phelps theorem does not hold for A u (c 0 , Z ′′ ) for a certain Banach space Z, while our Lindenstrauss theorem does. In order to obtain our results, we first handle their polynomial cases.2010 Mathematics Subject Classification. Primary: 47H60, 46G20. Secondary: 46B28, 46B20.

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“…Moreover, for a reflexive Banach space X, it is true for every Banach space Y [23], and this result is generalized to a Banach space X with the Radon-Nikodým property [8]. Very recently, this study has also been extended to non-linear mappings, such as multi-linear mappings, polynomials and holomorphic mappings [1,5,11,14,15,20,21].…”

Section: Introductionmentioning

confidence: 86%

Uniform Convexity and the Bishop–Phelps–Bollobás Property

Kim¹,

Lee²

2014

Can. j. math.

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Abstract.A new characterization of the uniform convexity of Banach space is obtained in the sense of Bishop-Phelps-Bollobás theorem. It is also proved that the couple of Banach spaces (X, Y ) has the Bishop-Phelps-Bollobás property for every Banach space Y when X is uniformly convex. As a corollary, we show that the BishopPhelps-Bollobás theorem holds for bilinear forms on ℓ p × ℓ q (1 < p, q < ∞).

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Denseness of Norm-Attaining Mappings on Banach Spaces

Cited by 7 publications

References 26 publications

On the polynomial Lindenstrauss theorem

On the polynomial Lindenstrauss theorem

Bounded Holomorphic Functions Attaining their Norms in the Bidual

Uniform Convexity and the Bishop–Phelps–Bollobás Property

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