Characterizations of norm--parallelism in spaces of continuous functions

In this paper, we consider the characterization of norm–parallelism problem in some classical Banach spaces. In particular, for two continuous functions $f,g$ on a compact Hausdorff space $K$, we show that $f$ is norm–parallel to $g$ if and only if there exists a probability measure (i.e. positive and of full measure equal to $1$) $\mu$ with its support contained in the norm attaining set $\{x\in K:\mathrm{\mid }f(x)\mathrm{\mid }=\mathrm{\parallel }f\mathrm{\parallel }\}$ such that $\big|\int_K \overline{f(x)}g(x)d\mu(x)\big| = |f|,|g|$.