Orthogonality and smooth points in \( C(K) \) and \( C_b(Ω) \)
For the usual norm on spaces \( C(K) \) and \( C_b(Ω) \) of all continuous functions on a compact Hausdorff space \( K \) (all bounded continuous functions on a locally compact Hausdorff space \( Ω \)), the following equalities are proved: \[ _t → 0^+ \|f+tg\|_C(K) - \|f\|_C(K) t = _x {z |f(z)| = \|f\| (e^-i f(x) g(x)). \] \[ _t → 0^+ \|f+tg\|_C_b(Ω) - \|f\|_C_b(Ω) t = _δ > 0 _x {z |f(z)| \|f\| - δ (e^-i f(x) g(x)). \] These equalities are used to characterize the orthogonality in the sense of James (Birkhoff) in spaces \( C(K) \) and \( C_b(Ω) \) as well as to give necessary and sufficient conditions for a point on the unit sphere to be a smooth point. , number=4, journal=Eurasian Mathematical Journal, author=Kečkić Dragoljub J., year=2012, month=Dec., pages=44–52