Orthogonality and smooth points in \(C(K)\) and \(C_b(\Omega)\) (Q2839014)

Let \(X\) be a normed space, \(x, y \in X\), and \(\mathbb C\) denote the field of complex numbers. Then \(y\) is said to be orthogonal to \(x\) (in the sense of Birkhoff and James) if for all \(\lambda, \mu \in \mathbb C\) we have \(\|\lambda x + \mu y\| \geq \|\lambda x\|\).NEWLINENEWLINELet \(C(K)\) be the space of continuous functions on a compact Hausdorff space \(K\) and \(C_b(\Omega)\) the space of bounded continuous functions on a locally compact Hausdorff space \(\Omega\).NEWLINENEWLINEIn the paper under review, the author proves the following two equalities: NEWLINE\[NEWLINE\lim_{t \to 0^+}\frac{\|f + tg\|_{C(K)} - \|f\|_{C(K)}}{t} = \max_{x \in \{z: |f(z)| = \|f\|\}}\text{Re}(e^{-i\arg f(x)}g(x)),NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\lim_{t \to 0^+}\frac{\|f + tg\|_{C_b(\Omega)} - \|f\|_{C_b(\Omega)}}{t} = \inf_{\delta > 0}\sup_{x \in \{z: |f(z)| = \|f\| - \delta\}}\text{Re}(e^{-i\arg f(x)}g(x)).NEWLINE\]NEWLINE These equalities are then used to characterize Birkhoff-James orthogonality in the spaces \(C(K)\) and \(C_b(\Omega)\).NEWLINENEWLINEAlso, from the first equality it is obtained: \(f \in C(K)\) is a smooth point of the sphere centered at \(0\) with radius \(\|f\|\) if and only if \(f\) attains its norm at a single point. From the second equality it is obtained: For a normal space \(\Omega\), \(f \in C_b(\Omega)\) is a smooth point of the sphere centered at \(0\) with radius \(\|f\|\) if and only if \(f\) attains its norm at a unique point and there is \(\delta > 0\) such that \(E_\delta = \{x \in \Omega: |f(x)| \geq \|f\| - \delta\}\) is a compact set.