Metric Spaces

Suppose X be a nonempty set. A function p: X × X →R is known as a metric provided for all x, y, and z in X,

1. p(x, y) > 0
2. p(x, y) = 0 , iff x = y
3. p(x, y) = p(y, x)
4. p(x, y) ≤ p(x, z) + p(z, y)

A metric space is made up of a nonempty set and a metric on the set.

The term “metric space” is frequently denoted (X, p). The triangle inequality for the metric is defined by property (iv). The set R of all real numbers with p(x, y) = | x – y | is the classic example of a metric space.

Introduction to Metric Spaces

Let us take a closer look at the various concepts associated with metric spaces in this article.

Normed Linear Spaces

A norm is a nonnegative real-valued function ||.|| on a linear space X for any u, v ∈ X and real number a,

1. || u || = 0 if and only if u = 0.
2. || u + v|| ≤ || u || + || v||
3. ||au|| = || a ll || u ||.

A normed linear space is a linear space that has a norm. A metric p on a linear space X is induced by a norm ||.|| on X by establishing

p(x, y) = || x-y || for all x,y ∈ X.

The triangle inequality for the norm is defined by property (ii).

Discrete Metric

The discrete metric p is established for any nonempty set X by assigning p(x, y) = 0 if x = y and p(x, y)=1 if x ≠ y.

Metric Subspaces

Let Y be a nonempty subset of X in a metric space (X, p). The constraint of p to Y × Y thus defines a metric on Y, which we refer to as a metric subspace. Hence, a metric space is a nonempty subset of Euclidean space, of an L^P(E) space,1 ≤ p ≤ ∞, and of C[a, b].

The preceding equivalence relationship between metrics on a set is helpful.

Relation 1:

If there are positive values c₁ and c₂ such that for all x₁, x₂∈ X, two metrics p and σ are said to be equal on a set X.

c₁.σ(x₁, x₂) ≤ P(x₁,x₂)≤ c₂. σ(x₁, x₂)

Relation 2:

Isometry is defined as a mapping f from a metric space (X, p) to a metric space (Y, σ) that maps X onto Y and for all x₁, x₂∈ X.

σ(f (x₁), f(x₂)) =P(x₁,x₂)

Open Sets, Closed Sets and Convergent Sequences

Many ideas explored in Euclidean and general normed linear spaces can be easily and effectively applied to general metric spaces. They aren’t reliant on a linear framework.

Open Set

Suppose (X, p) be a metric space. For a point x in X, and also r > 0, the set

B(x, r) ≡ {x’∈X I p(x’, x) <r}

is known as the open ball centered at x of radius r. A subset O of X is considered to be open if an open ball centered at x is included in O for every point x∈O. A neighbourhood of x for a point x ∈ X is an open set that includes x.

Closed Set

A point x ∈ X is called a point of closure of E for a subset E of a metric space X if every neighbourhood of x includes a point in E. The closure of E is the set of E’s points of closure and is represented by

We clearly have

Convergence Sequence

In metric space (X, p), a sequence {x_n} is said to converge to the point x∈X, given

i.e.,, for each ∈ > 0, there should be an index N such that ∀ n > N, p(x_n, x) < ∈.

The limit of the sequence is the point at which the sequence converges, and we typically write {x_n } → x to represent the convergence of {x_n } to x.

Continuous Mapping Between Metric Spaces

The natural generalization of continuity for real-valued functions of a real variable is as follows:

At the point x∈X provided for any sequence {x_n} in X, a mapping f from a metric space X to a metric space Y is also said to be continuous.

if {f_n} →x, then {f(x_n)} → f(x).

If the mapping f is continuous at every point in X, it is said to be continuous.

Solved Example on Metric Spaces

Example:

Given that X is a metric space, with the metric d. Define

Also, prove that d’ is a metric on X.

Solution:

It is clearly given that d’(x, y) = 0, if and only of d(x, y) = 0.

Thus, iff x = y.

Assume that b, c are the non-negative numbers. As

It means,

Now, add 1 to both sides of the inequality and rearrange the terms, we get

Which represents,

Such that a ≤ b+c, for any non-negative numbers a, b and c.

Hence, for x, y, z ∈ X, the non-negative numbers, such as a = d(x, y), b = d(x, z) and c = d(z, y) satisfies the condition a ≤ b+c for the metric d, by the triangle inequality.

Thus, the equation (1) provides the triangle inequality for

Hence, we can say that d’ is a metric on X.