Orthogonality and smooth points in \( C(K) \) and \( C_b(\Omega) \)

Abstract

For the usual norm on spaces \( C(K) \) and \( C_b(\Omega) \) of all continuous functions on a compact Hausdorff space \( K \) (all bounded continuous functions on a locally compact Hausdorff space \( \Omega \)), the following equalities are proved: \[ \lim_{t \to 0^+} \frac{\|f+tg\|_{C(K)} - \|f\|_{C(K)}}{t} = \max_{x \in \{z \mid |f(z)| = \|f\|\}} \Re\left(e^{-i \arg f(x)} g(x)\right). \] \[ \lim_{t \to 0^+} \frac{\|f+tg\|_{C_b(\Omega)} - \|f\|_{C_b(\Omega)}}{t} = \inf_{\delta > 0} \sup_{x \in \{z \mid |f(z)| \ge \|f\| - \delta\}} \Re\left(e^{-i \arg f(x)} g(x)\right). \] These equalities are used to characterize the orthogonality in the sense of James (Birkhoff) in spaces \( C(K) \) and \( C_b(\Omega) \) as well as to give necessary and sufficient conditions for a point on the unit sphere to be a smooth point.