Product ordered Hausdorff continuous $\mathbb {R}^{n}$-interval functions

Abstract

Vector-valued Sendov and Hausdorff continuity with intervals from a codomain of real Euclidean vector spaces isomorphic to $\mathbb {R}^{n}$ (including a separable real Hilbert space case) are constructed using product orders. This is a specialization of van der Walt’s vector-valued Hausdorff continuity. The Dedekind order completion of continuous functions under product orders follows. Product ordered Sendov continuous functions pass the same property on to their codomain components, and in finite dimensions this is an equivalence. Hausdorff continuity behaves similarly. New ways to transform interval functions and Sendov and Hausdorff continuous functions using bounded linear operators are defined. An interpretation of Hausdorff continuity that is essentially basis independent results: Hausdorff continuity belongs basis independently to finite Euclidean space, and therefore to multivariate real analysis. Infinite dimensional real separable Hilbert space is also considered, with interesting results relevant to the study of Hilbert space.