Characterizations of norm--parallelism in spaces of continuous functions

TL;DR

This paper characterizes the concept of norm--parallelism in spaces of continuous functions on compact spaces, linking it to the existence of specific probability measures supported on norm-attaining sets.

Contribution

It provides a new characterization of norm--parallelism in continuous function spaces using probability measures supported on norm-attaining sets.

Findings

Norm--parallelism is characterized by the existence of a probability measure with support in the norm-attaining set.

The characterization involves an integral condition relating the functions and the measure.

This result connects geometric properties of functions with measure-theoretic conditions.

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 $\{x\in K: \, |f(x)| = \|f\|\}$ such that $\big|\int_K \overline{f(x)}g(x)d\mu(x)\big| = \|f\|\,\|g\|$.