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We present a counterexample to the converse of results on approximate point spectra. A norm inequality in terms of a distance function is established for a quotient bounded operator on a locally convex Hausdorff space on the set of complex numbers.
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A. K. Gaur, Spectra in Banach and locally convex algebras. Port. Math. 64 (2007), no. 1, pp. 11–19DOI 10.4171/PM/1773