On almost isometric embedding from \(C(\Omega)\) into \(C_0(\Omega_0)\) (Q2581239)

The author first gives a proof of the well-known fact that the space \(c\) of convergent sequences isometrically embeds into \(C(\Omega)\) for any infinite compact set \(\Omega\) [see \textit{H. E. Lacey}, ``The isometric theory of classical Banach spaces'' (Grundlehren 208, Springer--Verlag, Berlin--Heidelberg--New York (1974; Zbl 0285.46024)] and then uses it to show that for \(0 < \varepsilon < \frac{1}{9}\) there is no \(\varepsilon\)-isometry between the spaces \(C(\Omega)\) and \(C_0(\Gamma)\) when \(\Gamma\) is an infinite discrete set and \(\Omega\) is an infinite compact set. The \(\varepsilon\) appearing in the statement of the theorem is taken to be less than \(\frac{1}{5}\).