What is a metric on a set?

In mathematics, a metric or distance function is a function that gives a distance between each pair of point elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set, but not all topologies can be generated by a metric.

What is the standard metric on R?

A metric space is a set X together with such a metric. The prototype: The set of real numbers R with the metric d(x, y) = |x – y|. This is what is called the usual metric on R. The complex numbers C with the metric d(z, w) = |z – w|.

What is the maximum metric?

In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension.

Is Empty set a metric space?

A metric space is formally defined as a pair . The empty set is not such a pair, so it is not a metric space in itself.

Is R2 a metric space?

Define d : R2 × R2 → R by d(x, y) = √ (x1 − y1)2 + (x2 − y2)2 x = (x1,x2), y = (y1,y2). Then d is a metric on R2, called the Euclidean, or ℓ2, metric. It corresponds to the usual notion of distance between points in the plane.

Is R complete with any metric?

Theorem: R is a complete metric space — i.e., every Cauchy sequence of real numbers converges.

What is the d2 metric?

(2) On Rn with the standard euclidean metric d = d2, the function f : Rn → Rn defined by f(x) = d(x,0)x is continuous. Page 8. 8. 1. METRIC SPACES.

What is the D Infinity metric?

The plane with the supremum or maximum metric d((x1 , y1), (x2 , y2)) = max(|x1 – x2|, |y1 – y2| ). It is often called the infinity metric d . These last examples turn out to be used a lot. To understand them it helps to look at the unit circles in each metric. That is the sets { x.

Is xy 2 a metric?

However, since we require d(x0,x0) = 0, any nonnegative function f(x, y) such that f(x0,x0) = 0 is a metric on X. (a) kd, Solution: First, note that if X has more than one point, then the zero func- tion cannot be a metric on X; this implies that k = 0.

What is the Supremum metric?

Let A be the set of all bounded real-valued functions f:S→R. Let d:A×A→R be the function defined as: ∀f,g∈A:d(f,g):=supx∈S|f(x)−g(x)| d is known as the supremum metric on A.

Is r n complete?

Definition Rn is a complete metric space. Every Cauchy sequence in Rn converges to a point of Rn. Definition Cauchy sequence.