1. If a metric space has the property that every Cauchy sequence converges, then the metric space is said to be complete. Name Notes of Metric Space Author Prof. Shahzad Ahmad Khan Send by Tahir Aziz Sitemap, Follow us on the metric space R. a) The interior of an open interval (a,b) is the interval itself. Software Introduction When we consider properties of a “reasonable” function, probably the first thing that comes to mind is that it exhibits continuity: the behavior of the function at a certain point is similar to the behavior of the function in a small neighborhood of the point. 4. These notes are very helpful to prepare a section of paper mostly called Topology in MSc for University of the Punjab and University of Sargodha. YouTube Channel In … Sitemap, Follow us on Theorem: (i) A convergent sequence is bounded. Let be a Cauchy sequence in the sequence of real numbers is a Cauchy sequence (check it!). 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. Example 1. The diameter of a set A is defined by d(A) := sup{ρ(x,y) : x,y ∈ A}. BSc Section 4. d(x,z) ≤ d(x,y)+d(y,z) To understand this concept, it is helpful to consider a few examples of what does and does not constitute a distance function for a metric space. The family Cof subsets of (X,d)defined in Definition 9.10 above satisfies the following four properties, and hence (X,C)is a topological space. BHATTI. Show that the real line is a metric space. CC Attribution-Noncommercial-Share Alike 4.0 International. Sequences 11 §2.1. Basic Probability Theory This is a reprint of a text first published by John Wiley and Sons in 1970. MSc Section, Past Papers Use Math 9A. Metric Spaces Then d is a metric on R. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for R with this absolute-value metric. For example, the real line is a complete metric space. Solution: For any x;y2X= R, the function d(x;y) = jx yjde nes a metric on X= R. It can be easily veri ed that the absolute value function satis es the MSc Section, Past Papers Home Matric Section Participate Twitter CHAPTER 3. Notes (not part of the course) 10 Chapter 2. This is known as the triangle inequality. Recall the absolute value of a real number: Ix' = Ix if x > 0 Observe that Discrete metric space is often used as (extremely useful) counterexamples to illustrate certain concepts. De nition 1.6. 1. Privacy & Cookies Policy This metric, called the discrete metric… Then for any $x,y\in X$, $$\left| {\,d(x,\,A)\, - \,d(y,\,A)\,} \right|\,\, \le \,\,d(x,\,y).$$. By a neighbourhood of a point, we mean an open set containing that point. Sequences in metric spaces 13 §2.3. c) The interior of the set of rational numbers Q is empty (cf. Matric Section Metric Spaces 1. Problems for Section 1.1 1. Mathematical Events A metric space (X;d) is a non-empty set Xand a function d: X X!R satisfying (1) For all x;y2X, d(x;y) 0 and d(x;y) = 0 if and only if x= y. Exercise 2.16). METRIC AND TOPOLOGICAL SPACES 3 1. Metric Spaces The following de nition introduces the most central concept in the course. Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. Story 2: On January 26, 2004 at Tokyo Disneyland's Space Mountain, an axle broke on a roller coaster train mid-ride, causing it to derail. 1 Chapter 1 Metric Spaces 1.1 Metric Space 1.1-1 Definition. These notes are collected, composed and corrected by Atiq ur Rehman, PhD. Report Abuse BSc Section Metric space solved examples or solution of metric space examples. 3. Think of the plane with its usual distance function as you read the de nition. In this video, I solved metric space examples on METRIC SPACE book by ZR. YouTube Channel A set Uˆ Xis called open if it contains a neighborhood of each of its Notes of Metric Spaces These notes are related to Section IV of B Course of Mathematics, paper B. For each x ∈ X = A, there is a sequence (x n) in A which converges to x. Then (x n) is a Cauchy sequence in X. Theorem: If $(X,d_1)$ and $\left(Y,d_2\right)$ are complete then $X\times Y$ is complete. b) The interior of the closed interval [0,1] is the open interval (0,1). Report Abuse The pair (X, d) is then called a metric space. 1. with the uniform metric is complete. PPSC FSc Section Since is a complete space, the sequence has a limit. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. Real Variables with Basic Metric Space Topology This is a reprint of a text first published by IEEE Press in 1993. Thus (f(x But (X, d) is neither a metric space nor a rectangular metric space. all metric spaces, saving us the labor of having to prove them over and over again each time we introduce a new class of spaces. Let (X,d) be a metric space and (Y,ρ) a complete metric space. For any space X, let d(x,y) = 0 if x = y and d(x,y) = 1 otherwise. Open Ball, closed ball, sphere and examples, Theorem: $f:(X,d)\to (Y,d')$ is continuous at $x_0\in X$ if and only if $f^{-1}(G)$ is open is. Mathematical Events Theorem: The space $l^{\infty}$ is complete. Sequences in R 11 §2.2. Distance in R 2 §1.2. 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 Show that (X,d 2) in Example 5 is a metric space. We are very thankful to Mr. Tahir Aziz for sending these notes. Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. Figure 3.3: The notion of the position vector to a point, P These are also helpful in BSc. These notes are related to Section IV of B Course of Mathematics, paper B. Then (X, d) is a b-rectangular metric space with coefficient s = 4 > 1. Z jf(x)g(x)jd 1 pAp Z jfjpd + 1 qBq Z jgjqd but Ap = R jfjpd and Bq = R jgjqd , so this is 1 kfkpkgkq kfgk1 1 p + 1 q = 1 kfgk1 kfkpkgkq I.1.1. METRIC SPACES AND SOME BASIC TOPOLOGY (ii) 1x 1y d x˛y + S ˘ S " d y˛x d x˛y e (symmetry), and (iii) 1x 1y 1z d x˛y˛z + S " d x˛z n d x˛y d y˛z e (triangleinequal-ity). Theorem: If $(x_n)$ is converges then limit of $(x_n)$ is unique. In this video, I solved metric space examples on METRIC SPACE book by ZR. R, metric spaces and Rn 1 §1.1. De ne f(x) = xp … If (X;d) is a metric space, p2X, and r>0, the open ball of radius raround pis B r(p) = fq2Xjd(p;q) 0. Show that (X,d) in Example 4 is a metric space. Chapter 1. Theorem (Cantor’s Intersection Theorem): A metric space (X,d) is complete if and only if every nested sequence of non-empty closed subset of X, whose diameter tends to zero, has a non-empty intersection. It is easy to check that satisfies properties .Ðß.Ñ .>> >1)-5) so is a metric space. Home We are very thankful to Mr. Tahir Aziz for sending these notes. In mathematics, a metric space … It is easy to verify that a normed vector space (V, k. k) is a metric space with the metric d (x, y) = k x-y k. An inner product (., .) Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Chapter 8 Euclidean Space and Metric Spaces 8.1 Structures on Euclidean Space 8.1.1 Vector and Metric Spaces The set K n of n -tuples x = ( x 1;x 2:::;xn) can be made into a vector space by introducing the standard operations of addition and scalar multiplication Example 1.1.2. A metric space is given by a set X and a distance function d : X ×X → R … A metric space is called complete if every Cauchy sequence converges to a limit. 94 7. If d(A) < ∞, then A is called a bounded set. 2. Theorem: $f:\left(X,d\right)\to\left(Y,d'\right)$ is continuous at $x_0\in X$ if and only if $x_n\to x$ implies $f(x_n)\to f(x_0)$. A point x2Xis a limit point of Uif every non-empty neighbourhood of x contains a point of U:(This de nition di ers from that given in Munkres). Already know: with the usual metric is a complete space. Software Theorem: A subspace of a complete metric space (, Theorem (Cantor’s Intersection Theorem): A metric space (. (iii)d(x, z) < d(x, y) + d(y, z) for all x, y, z E X. These are updated version of previous notes. 78 CHAPTER 3. Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) Metric space 2 §1.3. Let A be a dense subset of X and let f be a uniformly continuous from A into Y. In R2, draw a picture of the open ball of radius 1 around the origin in the metrics d 2, d 1, and d 1. Pointwise versus uniform convergence 18 §2.4. - [Lapidus] Wlog, let a;b<1 (otherwise, trivial). How to prove Young’s inequality. (ii) If $(x_n)$ converges to $x\in X$, then every subsequence $\left(x_{n_k}\right)$ also converges to $x\in X$. (y, x) = (x, y) for all x, y ∈ V ((conjugate) symmetry), 2. Twitter on V, is a map from V × V into R (or C) that satisfies 1. Theorem: The Euclidean space $\mathbb{R}^n$ is complete. These are actually based on the lectures delivered by Prof. Muhammad Ashfaq (Ex HoD, Department of Mathematics, Government College Sargodha). The definitions will provide us with a useful tool for more general applications of the notion of distance: Definition 1.1. Theorem: The union of two bounded set is bounded. The most important example is the set IR of real num- bers with the metric d(x, y) := Ix — yl. Proof. The cause was a part being the wrong size due to a conversion of the master plans in 1995 from English units to Metric units. A subset U of a metric space X is said to be open if it Let Xbe a linear space over K (=R or C). De¿nition 3.2.2 A metric space consists of a pair S˛d –a set, S, and a metric, d, on S. Remark 3.2.3 There are three commonly used (studied) metrics for the set UN. Report Error, About Us Example 7.4. metric space. We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. Step 1: define a function g: X → Y. Example 1.1.2. Since kx−yk≤kx−zk+kz−ykfor all x,y,z∈X, d(x,y) = kx−yk defines a metric in a normed space. FSc Section Neighbourhoods and open sets 6 §1.4. Theorem. Definition and examples Metric spaces generalize and clarify the notion of distance in the real line. Report Error, About Us Facebook Contributors, Except where otherwise noted, content on this wiki is licensed under the following license:CC Attribution-Noncommercial-Share Alike 4.0 International, Theorem: Let $(X,d)$ be a metric space. There are many ways. 3. Many mistakes and errors have been removed. Let f: X → X be defined as: f (x) = {1 4 if x ∈ A 1 5 if x ∈ B. A metric space is a pair ( X, d ), where X is a set and d is a metric on X; that is a function on X X such that for all x, y, z X, we De nition 1.1. Then f satisfies all conditions of Corollary 2.8 with ϕ (t) = 12 25 t and has a unique fixed point x = 1 4. Participate NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. BHATTI. 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