Metric spaces a metric space is a set x that has a notion of the distance dx,y between every pair of points x,y. A metric space is complete if every cauchy sequence has a limit. Some important properties of this idea are abstracted into. X r which measures the distance dx,y beween points x,y.
X y between metric spaces is continuous if and only if f. Variety of examples along with real life applications have been provided to understand and appreciate the beauty of metric spaces. Let us take a look at some examples of metric spaces. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. First, suppose f is continuous and let u be open in y. A metric space is called complete if every cauchy sequence converges to a limit. A metric space is a pair x, d, where x is a set and d is a.
In other words, no sequence may converge to two di. Turns out, these three definitions are essentially equivalent. Examples of metrics, elementary properties and new metrics from old ones problem 1. Real analysismetric spaces wikibooks, open books for an. I have included 295 completely worked out examples to illustrate and clarify all. Then f is continuous on x iff f 1o is an open set in x whenever o is an open set in y. A subset k of x is compact if every open cover of k has a. Moreover the concepts of metric subspace, metric superspace. The following standard lemma is often useful, and makes explicit a bit of intuition. Cauchy sequences and complete metric spaces lets rst consider two examples of convergent sequences in r. Some students, especially mathematically inclined ones, love these books, but others nd them hard to read. The observation above that the given metric on rn gives the usual notion of distance is what.
Uniform metric let be any set and let define particular cases. Here we can think of the fr as a copy of r living inside of r2. It helps to have a unifying framework for discussing both random variables and stochastic processes, as well as their convergence, and such a framework is provided by metric spaces. Metricandtopologicalspaces university of cambridge. The limit of a sequence in a metric space is unique. As we said, the standard example of a metric space is rn, and r, r2, and r3 in particular. Let aand bbe irrational numbers such that a 0, the open ball of radius. Show from rst principles that if v is a vector space over r or c then for any set xthe space 5. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur.
Let aand bbe irrational numbers such that a metric spaces of functions. Then d is a metric on r2, called the euclidean, or. Since every continuous function on a closed and bounded interval is bounded, therefore we have i i i i. Notes on metric spaces these notes are an alternative to the textbook, from and including closed sets and open sets page 58 to and excluding cantor sets page 95 1 the topology of metric spaces assume m is a metric space with distance function d. Properties of open subsets and a bit of set theory16 3. Jan 27, 2012 this video discusses an example of particular metric space that is complete. Wesaythatasequencex n n2n xisacauchy sequence ifforall0 thereexistsann. Definition a metric space is a set x together with a function d called a metric or distance function which assigns a real number d. A metric space is compact if and only if it is complete and totally bounded. Consider q as a metric space with the usual metric. Definition and fundamental properties of a metric space. Metric spaces and some basic topology ii 1x 1y d x. Vg is a linear space over the same eld, with pointwise operations. Then this does define a metric, in which no distinct pair of points are close.
You may have realised from your work on exercise 1. Informally, 3 and 4 say, respectively, that cis closed under. Mathematical proof or they may be 2place predicate symbols. The particular distance function must satisfy the following conditions. In particular, whenever we talk about the metric spaces rn without explicitly specifying the metrics, these are the ones we are talking about. Free and bound variables 3 make this explicit in each formula. Regrettably mathematical and statistical content in pdf files is unlikely to be accessible. Examples of compact metric spaces include the closed interval 0,1 with the absolute value metric, all metric spaces with finitely many points, and the cantor set. A metric space x is compact if every open cover of x has a. A metric space is a set x where we have a notion of distance. We then have the following fundamental theorem characterizing compact metric spaces.
The analogues of open intervals in general metric spaces are the following. Thetriangularinequalityis awellknownresultfromlinearalgebra,knownasthecauchyschwartzinequality. These observations lead to the notion of completion of a metric. If you are trying seriously to learn the subject, give them a look when you have the chance. Sometimes restrictions are indicated by use of special letters for the variables. In calculus on r, a fundamental role is played by those subsets of r which are intervals. For the theory to work, we need the function d to have properties similar to the distance.
A metric space x is sequentially compact if every sequence of points in x has a convergent subsequence converging to a point in x. Thus we could have solved the example of approximating ft t by using an or. It is also sometimes called a distance function or simply a distance. Ais a family of sets in cindexed by some index set a,then a o c. Introduction when we consider properties of a reasonable function, probably the. Show that the manhatten metric or the taxicab metric. Suppose x n is a convergent sequence which converges to two di. Metrics on spaces of functions these metrics are important for many of the applications in. This video discusses an example of particular metric space that is complete. The set of rational numbers q is a dense subset of r.
Often d is omitted and one just writes x for a metric space if it is clear from the context what metric is being used. A pair, where is a metric on is called a metric space. These will be the standard examples of metric spaces. For the theory to work, we need the function d to have properties similar. This is an isometry when r is given the usual metric and r2 is given the 2dimensional euclidean metric, but not. The following properties of a metric space are equivalent. As a formal logical statements, this theorem can be written in the following form. It turns the spotlight on the salient points of the theory, and shows what is the important questions. It is also sometimes called a distance function or simply a distance often d is omitted and one just writes x for a metric space if it is clear from the context what metric is being used we already know a few examples of metric spaces.
Lecture 3 complete metric spaces 1 complete metric spaces 1. Every closed subset of a compact space is itself compact. The fact that every pair is spread out is why this metric is called discrete. If is the real line with usual metric, then remarks. A particular case of the previous result, the case r 0, is that in every metric space singleton sets are closed. We do not develop their theory in detail, and we leave the veri. The metric is often regarded as a distance function. Also included are several worked examples and exercises.
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