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: lim t→0+ ||f + tg|| C(K) − ||f || C(K) t = max x∈{z | |f (z)|=||f ||} Re(e −i arg f (x) g(x)). and lim t→0+ ||f + tg|| C b (Ω) − ||f || C b (Ω) t = inf δ>0 sup x∈{z | |f (z)|≥||f ||−δ} Re(e −i arg 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.