On holomorphic functions attaining their norms |
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Affiliation: | a Departamento de Análisis Matemático, Universidad de Granada, 18071 Granada, Spain b Departamento de Análisis Matemático, Universidad de Valencia, Valencia, Spain |
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Abstract: | We show that on a complex Banach space X, the functions uniformly continuous on the closed unit ball and holomorphic on the open unit ball that attain their norms are dense provided that X has the Radon-Nikodym property. We also show that the same result holds for Banach spaces having a strengthened version of the approximation property but considering just functions which are also weakly uniformly continuous on the unit ball. We prove that there exists a polynomial such that for any fixed positive integer k, it cannot be approximated by norm attaining polynomials with degree less than k. For , a predual of a Lorentz sequence space, we prove that the product of two polynomials with degree less than or equal two attains its norm if, and only if, each polynomial attains its norm. |
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Keywords: | Holomorphic function Polynomial Norm attaining Lorentz sequence space |
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