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From Sets and Types to Topology and AnalysisTowards practicable foundations for constructive mathematics$
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Laura Crosilla and Peter Schuster

Print publication date: 2005

Print ISBN-13: 9780198566519

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198566519.001.0001

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APPROXIMATIONS TO THE NUMERICAL RANGE OF AN ELEMENT OF A BANACH ALGEBRA

APPROXIMATIONS TO THE NUMERICAL RANGE OF AN ELEMENT OF A BANACH ALGEBRA

Chapter:
(p.293) 19 APPROXIMATIONS TO THE NUMERICAL RANGE OF AN ELEMENT OF A BANACH ALGEBRA
Source:
From Sets and Types to Topology and Analysis
Author(s):

Douglas Bridges

Robin Havea

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566519.003.0019

The theory of Banach algebras presents a fine interplay between the topological and the algebraic, and as such is a good challenge for the constructive mathematician. In constructive mathematics, since the Hahn-Banach theorem does not produce norm-preserving extensions of linear functionals, the numerical range of an element of a Banach algebra has to be described using approximations. Nevertheless, these approximations suffice to produce constructive proofs of theorems such as that of Allan Sinclair on the spectral range of a Hermitian element. This chapter indicates once more that exact solutions whose existence can merely be guaranteed by classical logic are often unnecessary to prove a statement of a concrete character even when they occur in most classical proofs.

Keywords:   constructive mathematics, Hahn-Banach theorem, approximations, Sinclair theorem

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