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november 2019 Ordering among the topologies induced by various polynomial norms
L. Bernal-González, H.J.Cabana Méndez, G.A. Muñoz-Fernández, J.B. Seoane-Sepúlveda
Bull. Belg. Math. Soc. Simon Stevin 26(4): 481-492 (november 2019). DOI: 10.36045/bbms/1576206349

Abstract

Let us consider the following two norms in the vector space \,$\mathcal{P}$ \,of all complex polynomials: $$ \|p\|_{D_{r}}:= \sup\{|p(z)|: |z|<r\}, \text{ and } \|p\|_{1}:=\sum_{i=0}^{n}|a_{i}|, $$ where \,$p(z) = \sum_{i=0}^{n} a_{i} z^{i}$. In this note we show that, if $0 <\varepsilon < \varepsilon' < 1 < r < r'$, then $$ \|\cdot\|_{D_{\varepsilon}}\prec\|\cdot\|_{D_{\varepsilon'}}\prec\|\cdot\|_{D_{1}}\prec \|\cdot\|_{1}\prec \|\cdot\|_{D_{r}}\prec \|\cdot\|_{D_{r'}}, $$ where \,$\prec$ \,represents the natural (strict) partial order in their corresponding induced topologies.

Citation

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L. Bernal-González. H.J.Cabana Méndez. G.A. Muñoz-Fernández. J.B. Seoane-Sepúlveda. "Ordering among the topologies induced by various polynomial norms." Bull. Belg. Math. Soc. Simon Stevin 26 (4) 481 - 492, november 2019. https://doi.org/10.36045/bbms/1576206349

Information

Published: november 2019
First available in Project Euclid: 13 December 2019

zbMATH: 07167738
MathSciNet: MR4042395
Digital Object Identifier: 10.36045/bbms/1576206349

Subjects:
Primary: 54A10
Secondary: 30C10 , 46B50 , 47B38

Keywords: induced topology , polynomial norm , strict ordering

Rights: Copyright © 2019 The Belgian Mathematical Society

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Vol.26 • No. 4 • november 2019
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