APPROXIMATIONS TO THE NUMERICAL RANGE OF AN ELEMENT OF A BANACH ALGEBRA
Douglas Bridges, Robin Havea · 2005
Abstract 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.