Characterizations of norm--parallelism in spaces of continuous functions

Abstract

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 $\{x\in K:|f(x)|=\Vert f\Vert \}$ such that $|{\int }_K\overline{f(x)}g(x)d\mu (x)|=\Vert f\Vert \Vert g\Vert$.

arXiv commentto appear in Bulletin of the Iranian Mathematical Society