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On holomorphic functions attaining their norms. (English) Zbl 1086.46034
Authors’ 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 attaining their norms are dense provided that $X$ has the Radon-Nikodym property. We also show that the same result holds for Banach spaces having the strengthened version of the approximation property but considering just functions with 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 $X = d_{w^*}(w,1),$ a predual of a Lorentz sequence space, we prove that the product of two polynomials with degree less than or equal to two attains its norm if, and only if, each polynomial attains its norm.

##### MSC:
 46G25 (Spaces of) multilinear mappings, polynomials 46B25 Classical Banach spaces in the general theory of normed spaces 46B03 Isomorphic theory (including renorming) of Banach spaces 46G20 Infinite dimensional holomorphy
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