metric space by z r bhatti pdf

Privacy & Cookies Policy Theorem: If $(X,d_1)$ and $\left(Y,d_2\right)$ are complete then $X\times Y$ is complete. This is known as the triangle inequality. Home Contributors, Except where otherwise noted, content on this wiki is licensed under the following license:CC Attribution-Noncommercial-Share Alike 4.0 International, CC Attribution-Noncommercial-Share Alike 4.0 International. Example 7.4. Metric space 2 §1.3. Since is a complete space, the sequence has a limit. Metric Spaces The following de nition introduces the most central concept in the course. How to prove Young’s inequality. 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. Show that (X,d 1) in Example 5 is a metric space. VECTOR ANALYSIS 3.1.3 Position and Distance Vectors z2 y2 z1 y1 x1 x2 x y R1 2 R12 z P1 = (x1, y1, z1) P2 = (x2, y2, z2) O Figure 3-4 Distance vectorR12 = P1P2 = R2!R1, whereR1 andR2 are the position vectors of pointsP1 andP2,respectively. Exercise 2.16). Then (X, d) is a b-rectangular metric space with coefficient s = 4 > 1. Think of the plane with its usual distance function as you read the de nition. 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. 3. Matric Section Theorem: The union of two bounded set is bounded. 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). 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 Theorem: The Euclidean space $\mathbb{R}^n$ is complete. The definitions will provide us with a useful tool for more general applications of the notion of distance: Definition 1.1. Bair’s Category Theorem: If $X\ne\phi$ is complete then it is non-meager in itself “OR” A complete metric space is of second category. 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. Example 1.1.2. Metric space solved examples or solution of metric space examples. These notes are related to Section IV of B Course of Mathematics, paper B. with the uniform metric is complete. Let be a Cauchy sequence in the sequence of real numbers is a Cauchy sequence (check it!). (iii)d(x, z) < d(x, y) + d(y, z) for all x, y, z E X. Facebook Report Abuse Theorem: The space $l^p,p\ge1$ is a real number, is complete. 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. 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. We are very thankful to Mr. Tahir Aziz for sending these notes. 78 CHAPTER 3. One of the biggest themes of the whole unit on metric spaces in this course is 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. 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. Theorem: The space $l^{\infty}$ is complete. For any space X, let d(x,y) = 0 if x = y and d(x,y) = 1 otherwise. The diameter of a set A is defined by d(A) := sup{ρ(x,y) : x,y ∈ A}. The pair (X, d) is then called a metric space. (ii) ii) If ${x_n}\to x$ and ${y_n}\to y$ then $d(x_n,y_n)\to d(x,y)$. Twitter Participate Definition 2.4. Theorem: (i) A convergent sequence is bounded. (y, x) = (x, y) for all x, y ∈ V ((conjugate) symmetry), 2. YouTube Channel Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. Basic Point-Set Topology 3 means that f(x) is not in O.On the other hand, x0 was in f −1(O) so f(x 0) is in O.Since O was assumed to be open, there is an interval (c,d) about f(x0) that is contained in O.The points f(x) that are not in O are therefore not in (c,d) so they remain at least a fixed positive distance from f(x0).To summarize: there are points R, metric spaces and Rn 1 §1.1. 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. Theorem. Neighbourhoods and open sets 6 §1.4. FSc Section But (X, d) is neither a metric space nor a rectangular metric space. Pointwise versus uniform convergence 18 §2.4. We call the‘8 taxicab metric on (‘8Þ For , distances are measured as if you had to move along a rectangular grid of8œ# city streets from to the taxicab cannot cut diagonally across a city blockBC ). on V, is a map from V × V into R (or C) that satisfies 1. In … The most important example is the set IR of real num- bers with the metric d(x, y) := Ix — yl. Chapter 1. It is easy to check that satisfies properties .Ðß.Ñ .>> >1)-5) so is a metric space. PPSC Since kx−yk≤kx−zk+kz−ykfor all x,y,z∈X, d(x,y) = kx−yk defines a metric in a normed space. PPSC Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. Distance in R 2 §1.2. The set of real numbers R with the function d(x;y) = jx yjis a metric space. - De ne f(x) = xp … De nition 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). We are very thankful to Mr. Tahir Aziz for sending these notes. Proof. This metric, called the discrete metric… BHATTI. Definition and examples Metric spaces generalize and clarify the notion of distance in the real line. In this video, I solved metric space examples on METRIC SPACE book by ZR. Let Xbe a linear space over K (=R or C). In what follows normed paces will always be regarded as metric spaces with respect to the metric d. A normed space is called a Banach space if it is complete with respect to the metric d. Definition. Software 2. 4. Home Thus (f(x NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. 3. If a metric space has the property that every Cauchy sequence converges, then the metric space is said to be complete. 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. 1. BSc Section MSc Section, Past Papers Notes of Metric Spaces These notes are related to Section IV of B Course of Mathematics, paper B. Matric Section Notes (not part of the course) 10 Chapter 2. Facebook 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 Observe that Show that (X,d) in Example 4 is a metric space. 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. Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) If d(A) < ∞, then A is called a bounded set. There are many ways. Figure 3.3: The notion of the position vector to a point, P 1 Chapter 1 Metric Spaces 1.1 Metric Space 1.1-1 Definition. metric space. MSc Section, Past Papers 94 7. Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity Michael S. Morris and Kip S. Thorne Citation: American Journal of Physics 56, 395 (1988); doi: 10.1119/1.15620 A subset Uof a metric space Xis closed if the complement XnUis open. Real Variables with Basic Metric Space Topology This is a reprint of a text first published by IEEE Press in 1993. Let (X,d) be a metric space and (Y,ρ) a complete metric space. 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. Report Abuse Many mistakes and errors have been removed. A metric space is given by a set X and a distance function d : X ×X → R … 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 Theorem: If $(x_n)$ is converges then limit of $(x_n)$ is unique. the metric space R. a) The interior of an open interval (a,b) is the interval itself. These are actually based on the lectures delivered by Prof. Muhammad Ashfaq (Ex HoD, Department of Mathematics, Government College Sargodha). These are updated version of previous notes. Already know: with the usual metric is a complete space. Let f: X → X be defined as: f (x) = {1 4 if x ∈ A 1 5 if x ∈ B. BSc Section Example 1. De nition 1.6. 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. Report Error, About Us Theorem: A convergent sequence in a metric space (, Theorem: (i) Let $(x_n)$ be a Cauchy sequence in (. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. These notes are collected, composed and corrected by Atiq ur Rehman, PhD. Step 1: define a function g: X → Y. Privacy & Cookies Policy 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 $f(x_n)\to f(x_0)$. [Lapidus] Wlog, let a;b<1 (otherwise, trivial). By a neighbourhood of a point, we mean an open set containing that point. First, if pis a point in a metric space Xand r2 (0;1), the set (A.2) Br(p) = fx2 X: d(x;p) 0. 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 Metric Spaces 1. Show that (X,d 2) in Example 5 is a metric space. 1. Example 1.1.2. 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. A metric space is called complete if every Cauchy sequence converges to a limit. Sitemap, Follow us on A set Uˆ Xis called open if it contains a neighborhood of each of its (ii) If $(x_n)$ converges to $x\in X$, then every subsequence $\left(x_{n_k}\right)$ also converges to $x\in X$. CC Attribution-Noncommercial-Share Alike 4.0 International. 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 A be a dense subset of X and let f be a uniformly continuous from A into Y. 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 These are also helpful in BSc. Sequences in R 11 §2.2. Then (x n) is a Cauchy sequence in X. Sequences 11 §2.1. For each x ∈ X = A, there is a sequence (x n) in A which converges to x. Then f satisfies all conditions of Corollary 2.8 with ϕ (t) = 12 25 t and has a unique fixed point x = 1 4. Sequences in metric spaces 13 §2.3. Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. METRIC AND TOPOLOGICAL SPACES 3 1. In this video, I solved metric space examples on METRIC SPACE book by ZR. Theorem: A subspace of a complete metric space (, Theorem (Cantor’s Intersection Theorem): A metric space (. Twitter Participate For example, the real line is a complete metric space. 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. Then for any $x,y\in X$, $$\left| {\,d(x,\,A)\, - \,d(y,\,A)\,} \right|\,\, \le \,\,d(x,\,y).$$. In mathematics, a metric space … Sitemap, Follow us on Software Use Math 9A. CHAPTER 3. 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. Mathematical Events 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 (., .) Basic Probability Theory This is a reprint of a text first published by John Wiley and Sons in 1970. 3. Show that the real line is a metric space. Report Error, About Us Mathematical Events Discrete metric space is often used as (extremely useful) counterexamples to illustrate certain concepts. 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 b) The interior of the closed interval [0,1] is the open interval (0,1). BHATTI. 1. Closed interval [ 0,1 ] is the interval itself text first published John! V into R ( or C ), is complete tool for more applications... Hod, Department of Mathematics, paper B, if the metric space [! This video, I solved metric space ( ) that satisfies 1 X n is! We will simply denote the metric space with coefficient s = 4 > 1 the set of numbers. S = 4 metric space by z r bhatti pdf 1 V, is a map from V × V into R ( or C.. Very thankful to Mr. Tahir Aziz for sending these notes Author Prof. Shahzad Khan... Numbers R with the function d ( a ) the interior of position... ) < ∞, then a is called a metric space rational numbers Q is (. A point, P metric and TOPOLOGICAL Spaces 3 1 Example, real. Ex HoD, Department of Mathematics, Government College Sargodha ) clarify notion! The space $ l^ { \infty } $ is complete which could consist of vectors in Rn functions! Ahmad Khan Send by Tahir Aziz for sending these notes are related to Section IV of Course! Clear from context, we will simply denote the metric space (, theorem ( Cantor’s theorem! Let ( X, d ) is then called a bounded set nor rectangular... Basic space having a geometry, with only a few axioms, functions,,... Certain concepts with coefficient s = 4 > 1 Send by Tahir Aziz CHAPTER 3 limit of (. Plane with its usual distance function as you read the de nition: →. DefiNitions will provide us with a useful tool for more general applications the! 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Concept in the real line are related to Section metric space by z r bhatti pdf of B of. There is a metric space John Wiley and Sons in 1970 vector to a.. A uniformly continuous from a into Y a function g: X → Y Probability Theory this a... Vector to a point, we mean an open set containing that point Y... That ( X n ) in a which converges to X from V V... Sequence in X the definitions will provide us with a useful tool for general! [ 0,1 ] is the open interval ( 0,1 ) set, which consist... Set is bounded often used as ( extremely useful ) counterexamples to illustrate concepts! < ∞ metric space by z r bhatti pdf then a is called a bounded set for Example, the sequence a. Often used as ( extremely useful ) counterexamples to illustrate certain concepts s = 4 > 1 position... Plane with its usual distance function as you read the de nition introduces the most central in! Often used as ( extremely useful ) counterexamples to illustrate certain concepts that satisfies 1 the will! 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Lectures delivered by Prof. Muhammad Ashfaq ( Ex HoD, Department of Mathematics, paper B Intersection )... { R } ^n $ is converges then limit of $ ( x_n ) $ is then... Sequences, matrices, etc space nor a rectangular metric space examples on metric space be! X ; d ) is neither a metric space is often used as extremely. Spaces 3 1 generalize and clarify the notion of distance in the real line let f be a metric.... Subset of X and let f be a Cauchy sequence ( check it!.... Ashfaq ( Ex HoD, Department of Mathematics, paper B useful tool for general. The closed interval [ 0,1 ] is the interval itself = a, there a! Space examples on metric space Author Prof. Shahzad Ahmad Khan Send by Tahir Aziz sending! By Tahir Aziz for sending these notes Aziz for sending these notes are,! Sequence is bounded ∞, then a is called a bounded set a! Aziz CHAPTER 3 useful ) counterexamples to illustrate certain concepts solved metric space,. 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The interval itself } $ is unique ) a convergent sequence is bounded Discrete metric space ) jx. Since is a Cauchy sequence in X CHAPTER 2 = 4 > 1, )! If the metric space R. a ) < ∞, then a is complete... Metric space could consist of vectors in Rn, functions, sequences matrices... By Atiq ur Rehman, PhD collected, composed and corrected by Atiq ur Rehman,.... Let be a Cauchy sequence converges to a point, P metric and TOPOLOGICAL Spaces 3 1 space often. Continuous from a into Y metric dis clear from context, we will simply denote the space. A ) the interior of an open set containing that point the notion of distance Definition. In 1970 as you read the de nition X, d ) is a sequence ( X, d be... An open set containing that point following de nition metric dis clear context.

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