6533b82cfe1ef96bd128eb78
RESEARCH PRODUCT
On holomorphic functions attaining their norms
María D. AcostaManuel MaestreDomingo GarcíaJ. Alaminossubject
Unit spherePure mathematicsMathematics::Functional AnalysisLorentz sequence spaceFunction spaceApproximation propertyApplied MathematicsMathematical analysisBanach spaceHolomorphic functionNorm attainingHolomorphic functionPolynomialUniform continuityNorm (mathematics)Ball (mathematics)AnalysisMathematicsdescription
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 X=d ∗ (ω,1) , 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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2004-09-01 | Journal of Mathematical Analysis and Applications |