Metric Space refers to a set in mathematics, and the distance between any elements in the set is definable.
Also called distance space.A special classtopological space 。Frechet (Fr é chet, M. - R.)Euclidean spaceThe concept of distance was abstracted and metric space was defined in 1906.
In metric space, compactness, countable compactness, sequential compactness and subset compactness are consistent.Separability, genetic separability, second countability and Lindlerf property are consistent.Metric space must satisfy the first countable axiom, which is Hausdorff space, completely normal space, paracompact space.Pseudometric spaces satisfy the first countable axiom, but are generally notHausdorff space。[1]
D is called a measure (or distance) of the set X.Call even pair (X, d) ametric space, or X is a metric space for metric d.
Concept introduction
Announce
edit
Related books
A basic, important, and closest thing toEuclidean spaceThe abstract space of.At the end of the 19th century, German mathematician Gcantor Set theory was founded, which laid the foundation for the establishment of various abstract spaces.In the early 20th century, French mathematician MR.Frechet Found manyAnalyticsFrom a more abstract point of view, the results of are all related to the distance relationship between functions, thus abstracting the concept of metric space.
In the metric space, the most consistent with our intuitive understanding of reality is three-dimensionaleuclidean space 。The Euclidean metric in this space defines the distance between two points as the length of the line segment connecting the two points.[2]
Detailed definition
Announce
edit
Metric space is also called distance space.one kindtopological space The topology on it is determined by the distance.Let R be a nonempty set and ρ (x, y) be a binary function on R, satisfying the following conditions:
Then ρ (x, y) is the distance between two points x, y, R becomes a metric space or a metric space according to the distance ρ, and is recorded as (R, ρ). Let A be a subset of R, then A also becomes a metric space according to the distance ρ in R, and is called the (metric) subspace of R. If the condition 1 of the above distance is changed to ρ (x, y) ≥ 0 and ρ (x, x)=0, then ρ is called a quasi distance on R. When ρ (x, y)=0, it is recorded as x~yIs an equivalence relation on R, the quotient set (i.e. all equivalence classes) is D=R/～, and the binary function ρ ～ is made on D: ρ ～ (x ～, y ～)=ρ (x, y) (x ∈ x ～, y ∈ y ～), then ρ ～ is the distance on D, and (D, ρ ～) is called the quotient (metric) space derived by R according to quasi distance ρ
The subset A in the metric space (R, ρ) is called bounded if xzero∈ R, there is a constant M, so that ρ (xzero, x) ≤ M holds for all x in A. Let xzero∈ R, r>0, then the set {x | x ∈ R, ρ (x, xzero）<r} is taken as xzeroIs the center, r is the radius of the tee, or xzeroR neighborhood of, denoted as O (xzero, r). Let A ⊂ R. If for any x ∈ A, there exists a neighborhood O (x, r) ⊂ A of x, then A is called an open set;The complement of an open set is called a closed setThe smallest closed set containing subset A in R is called A'sclosure。
The metric space isFrechet (Fr é chet, M. - R.) was introduced in 1906. It is a basic and important abstract space in modern mathematics and very close to Euclidean spacefunctional analysisOne of the foundations of.
Related concepts
Announce
edit
Of subset S of metric space MclosureCl (S) is the smallest of M containing SClosed set。
If any hollow tee of a point x in metric space M contains at least one point in S, then x is called S'sLimit pointorAccumulation point。
The subset S of metric space M is calledDense subset, if its closure is M.If any tee of a point x in M contains at least one point in S, S is a dense subset.
The metric space is calledSeparable metric space, if it contains countable dense subsets.[5]
Basic examples
Announce
edit
Let X be any non empty set, and define the mapping d: X × X → R as follows
⑴ For any element x in X, d (x, x)=0;
⑵ If x and y are two different elements in X, then d (x, y)=1
Then d thus defined satisfies (I) (II) (III) and is a measure of the set X.Such a measure is calleddiscrete metric。
limit
Prove that the limit of convergence sequence in metric space is unique[3]
Let {a_n} converge to a and to b.Then for any u>0, there is N so that for n>N there is d (a_n, a)<u/2 and d (a_n, b)<u/2, so d (a, b)<=d (a_n, a)+d (a_n, b)<u/2+u/2=u. d (a, b) is a non negative constant and less than any positive number u>0, so there must be d (a, b)=0, so a=b
euclidean space
The set Rn is composed of all n-ary real arrays (x1, x2,..., xn), where the element x=(x1, x2,..., xn)
The distance from y=(y1, y2,..., yn) is defined as
metric space
Space H
Where R represents a set of real numbers.Define the distance between elements x=(x1x2,..., xn,...) and y=(y1, y2,..., yn...) as
B={(x1, x2,..., xn,...) │ (xn ∈ R, n=1,2,...)} For two different elements x=(x1, x2,..., xn,...) and y=(y1,y2，…，yn，…），Use m (x, y) to represent the minimum label n satisfying xn ≠ yn, and define the distance between x and y as:
Then specify d (x, x)=0 (x ∈ B).Generally, suppose Ω is any set, take X={(x1, x2,... xn,...) | xn ∈ Ω), and define m (x, y) and d (x, y) in the same way. The resulting metric space is also called Bell space.
function space
When dealing with analysis problems, various function spaces can be introduced according to specific needs.For example, consider definingClosed intervalThe set of all continuous real valued functions on [0,1] can define the distance between two functions ƒ and g as
metric space
For metric space X, we can use its metric d to introduce a topological structure whose base element is all the kickouts B (x, r)={y ∈ X | d (x, y)<r}. This topological structure is called metric d;On the same set, different metrics can produce the same topology.
The same topology is generated.Measure is nottopologyConcept.
Dense subspace
metric space
In metric space, the convergence concept of point sequence can be defined by distance: xn→xzeroIt refers to d (xn,xzero)。Point column {xn}It is called Cauchy point sequence, which means thatPositive real numberε，There is a natural number N, so when m, n ≥ N, it can be proved that the convergence point sequence must be Cauchy point sequence, and the reverse is not true.The metric space in which every Cauchy point sequence converges is calledComplete metric space。This kind of space has many good properties.For example, the principle of contractive mapping in a complete metric space holds.It can be used to prove differential equationsintegral equation And a series of existence and uniqueness theorems of infinite linear algebraic equations.Any subset Y of the metric space X plus the original distance also becomes the metric space, called XSubspace。If every tee {x ∈ X | d (x0, x)<r} contains Y points, then Y is said to be a dense subspace of X.
theorem
Announce
edit
Each metric space X is a dense subspace of another complete metric space, and is uniquely constructed by X.For example, a real line isRational number setIt is based on this important fact that the rigorous mathematical analysis theory was established at the beginning of the 20th century.
It can be proved that the intersection of countable dense open subsets in a complete metric space is stillDense and dense。[4]
topological space
Announce
edit
Metric space has many good properties, for example, it satisfies the first countable axiom, it is a Hausdorff space, a normal space, or a paracompact space.In addition, for metric spaces, compactness is equivalent to any of the following three: ① any countable open cover has finite sub cover; ②Every infinitysubsetAll in spaceAccumulation point: ③ Each point column has convergence sub columns.
Onetopological space Under what conditions can the topology of a metric space be used as the topology of a metric space?This is an important problem in point set topology theory, which is called the metrization problem.The two main results of the metrization problem are Urysohn metrization theorem, that is, every second countable normal Hausdorff space is metrizable (usually introduced in the course of point set topology), and Bing Nagata Smirnov metrization theorem, that istopological space Metrizableif and only ifIt is a regular Hausdorff space and has a countable locally finite base.
