Making given metric space bounded

Thread starter jostpuur
Start date May 18, 2008
Tags

Bounded Metric Metric space Space

In summary, a metric space is a mathematical concept that describes the distance between any two points in a given set. To make a metric space bounded, an upper bound for all distances between points must be found. This is important in defining the size and structure of the space and allows for the understanding of concepts like convergence and completeness. Common techniques for making a metric space bounded include the triangle inequality, compactness, and the Hausdorff metric. However, not all metric spaces can be made bounded, particularly infinite-dimensional spaces or spaces with unbounded distances. In these cases, alternative methods or concepts may be used to understand the space.

May 18, 2008

jostpuur

I just encountered a claim, that for any given metric space (X,d), there exists another topologically equivalent metric d' so that (X,d') is bounded. Anyone knowing anything about the proof for this?

Physics news on Phys.org

May 18, 2008

morphism

Science Advisor

Homework Helper

Try putting d'=d/(1+d) or min{1,d}.

May 18, 2008

jostpuur

Ok. I checked the min{1,d} one.

1. What is a metric space?

A metric space is a mathematical concept used to describe the distance between any two points in a given set. It is defined by a distance function or metric that satisfies certain properties, such as being non-negative and symmetric.

2. How do you make a given metric space bounded?

To make a given metric space bounded, you need to find a value that serves as an upper bound for all distances between any two points in the space. This can be done by finding the maximum distance between any two points or by using a specific value as the upper bound, such as the diameter of the space.

3. Why is it important to make a metric space bounded?

Making a metric space bounded is important because it helps to define the size and structure of the space. A bounded metric space allows for the definition of concepts such as convergence and completeness, which are essential in understanding the properties of the space and its elements.

4. What are some common techniques for making a metric space bounded?

There are several techniques for making a metric space bounded, including using the triangle inequality, compactness, and the Hausdorff metric. These techniques involve finding a value or function that bounds the distances between points in the space, ultimately defining the size and structure of the space.

5. Can all metric spaces be made bounded?

No, not all metric spaces can be made bounded. For example, infinite-dimensional spaces or spaces with unbounded distances cannot be made bounded. Additionally, some spaces may not have a well-defined upper bound for distances between points. In these cases, alternative methods or concepts may be used to understand the size and structure of the space.