What definition of equivalence for metrics are you using? The usual definition is that two metrics are equivalent if convergence of a sequence in one metric implies convergence in the other metric. That does not imply the condition $(1/c)T(u,v) \leqslant p(u,v) \leqslant cT(u,v)$ (for all $u,v\in X$), which is usually called strong equivalence of the two metrics (see the discussion in the Wikipedia page that I linked to above).wonguyen1995 said:Show that two metrics p and T on the same set X are equivalent if and only if there is a c > 0
such that for all u,v belong to X,
(1/c)T(u,v)=<p(u,v)=<cT(u,v)
Please help me , I'm so confused about Real Analysis.
A metric space is a mathematical structure that defines the distance between any two points in a set. It consists of a set of elements and a distance function, or metric, that satisfies certain properties such as non-negativity, symmetry, and the triangle inequality.
A metric space is a more general concept than a normed vector space. While both involve a distance function, a normed vector space also has the additional structure of vectors and operations such as addition and scalar multiplication. In contrast, a metric space can be any set of elements, and the distance function does not necessarily have to satisfy the conditions of a norm.
Metric spaces are essential in real analysis as they provide a framework for studying and analyzing the properties of continuous functions and their limits. They also allow for the development of important concepts such as convergence, completeness, and compactness, which are crucial in many areas of mathematics, including calculus and differential equations.
In a metric space, convergence of a sequence can be shown by proving that the distance between the terms of the sequence and the limit approaches zero as the index of the sequence gets larger. This can be done using the definition of convergence, the triangle inequality, and the properties of the distance function in the metric space.
No, not all sets can be made into a metric space. For a set to be a metric space, the distance function must satisfy certain properties, such as non-negativity, symmetry, and the triangle inequality. If these conditions are not met, then the set cannot be made into a metric space.