Abstract
The main goal of this chapter is to set the stage for the rest of this monograph by presenting a brief survey of some of the many facets of the theory of quasi-metric spaces. Quasi-metric spaces constitute generalizations of not only the classical Euclidean setting, but of quasi-Banach spaces and ultrametric spaces. In this work, quasi-metric spaces will constitute the natural geometric context in which our main results are going to be developed.
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Notes
- 1.
A function d: X → [0, ∞) shall be referred to as a distance provided for every x, y, z ∈ X, the function d satisfies: \(d(x,y) = 0 \Leftrightarrow x = y\), d(x, y) = d(y, x), and d(x, y) ≤ d(x, z) + d(z, y).
- 2.
Given a vector space \(\mathcal{X}\) over \(\mathbb{C}\), recall that a function \(\|\cdot \|: \mathcal{X} \rightarrow [0,\infty )\) is called a semi-norm provided that for each \(x,y \in \mathcal{X}\) the following three conditions hold (i) x = 0 implies \(\|x\| = 0\), (ii) \(\|\lambda x\| = \vert \lambda \vert \!\cdot \!\| x\|\), \(\forall \,\lambda \in \mathbb{C}\), and (iii) \(\|x + y\| \leq \| x\| +\| y\|\).
- 3.
Such points have been historically referred to as “atoms”.
- 4.
In general, given a nonempty set X, call a function μ: 2X → [0, ∞] an outer-measure if μ(∅) = 0 and \(\mu (E) \leq \sum _{j\in \mathbb{N}}\mu (E_{j})\) whenever \(E,\{E_{j}\}_{j\in \mathbb{N}} \subseteq 2^{X}\) satisfy \(E \subseteq \cup _{j\in \mathbb{N}}E_{j}\).
- 5.
Recall that given two arbitrary quasi-metric spaces (X j , q j ), j = 0, 1, a mapping Φ: (X 0 , q 0 ) → (X 1 , q 1) is called bi-Lipschitz provided for some (hence, any) ρj ∈ q j , j = 0, 1, one has ρ 1 (Φ(x),Φ(y)) ≈ρ 0 (x,y), uniformly for x,y ∈ X 0.
- 6.
- 7.
Call a quasi-metric space (X, ρ) pathwise connected provided for every pair of points x, y ∈ X, there exists a continuous path f: [0, 1] → (X, τ ρ ) with f(0) = x and f(1) = y, where τ ρ represents the canonical topology induced by the quasi-distance ρ on X. We shall refer to the set \(\Gamma:= f\big([0, 1]\big) \subseteq X\) as a continuous path joining x and y.
- 8.
In general, call (X, q, μ) a d-Ahlfors-regular ultrametric space for some d ∈ (0, ∞) if (X, q, μ) is a d- AR space and q contains an ultrametric.
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Alvarado, R., Mitrea, M. (2015). Geometry of Quasi-Metric Spaces. In: Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces. Lecture Notes in Mathematics, vol 2142. Springer, Cham. https://doi.org/10.1007/978-3-319-18132-5_2
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