metric space problems and solution pdf

Every convergent sequence in a metric space is a Cauchy sequence. A solutions manual for Topology by James Munkres - 9beach 18 Optimize Gift Card Spending Problem: Given gift cards to different stores and a shopping list of desired purchases, decide how to spend the gift cards to use as much of the gift card money as possible. Firstly, by virtue of Gerstewitz scalarization functions and oriented distance functions, a new scalarization function $$\\omega $$ ω is constructed and some properties of it are given. Let X be a topological space and let (Y,d) be a metric space. (PDF) INTRODUCTION TO METRIC SPACES - ResearchGate 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 a solution), the Theorem of the Maximum, (which tells us how the solution to a maximization problem will change with the parameters of that problem), and some fixed point theorems (which are used in proving that equilibria of certain systems exist). (b)Show that (X;d) is a complete metric space. xn → x. Show that the real line is a metric space. Section 8.2 discusses compactness in a metric space, and Sec- Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. However, such an embedding is not required to define the tangent space of a manifold (Walk 1984). Let 0 . Let be a Cauchy sequence in the sequence of real numbers is a Cauchy sequence (check it!). The class will take an abstract approach, especially around metric spaces and related concepts. 3. Let Aand Bbe compact subsets of a metric space (X;d). 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. Continuous map- We need one more lemma before proving the classical version of Ascoli's Theorem. (c) Generalize this to show that for any metric space (X;d);there is a bounded metric (i.e., one for which there exists M>0 such that the distance between any two points is less than C) that generates the same . 74 CHAPTER 3. Definition 2.5 A topological space is called if there exists aÐ\ß Ñg pseudometrizable pseudometric on such that If is a metric, then is called .\ œÞ . PDF FUNCTIONAL ANALYSIS - University of Pittsburgh Let U= fU ig i2I be an open cover of A[B. Exercises 2.1Show that the binary relation ˘on C[E] de ned above is an equivalence relation. the complete metric space K is a set of functions, and the map F transforms a function into another function (we often say that F is an operator ). constitute a distance function for a metric space. 1.1 Definition of a Metric To begin with we need to define a metric. More For some students, Math 115 may be a suitable . Suppose {x n} is a Cauchy sequence of points in A . Role of metrics in geometry and topology 48 3.6. 4. A rather trivial example of a metric on any set X is the discrete metric d(x,y) = {0 if x . These notes are also useful in the preparation of JAM, CSIR-NET, GATE, SET, NBHM, TIFR, …etc. Problems { Chapter 1 Problem 5.1. that an optimal solution can be viewed as a metric. 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. Convert the measurement to centigrams. 2.2Show that a ˘b if and only if D(a;b) = 0. Spaces of closed subsets of a compact metric space 57 3.10. Denote by Athe closure of A in X, and equip Y with the subspace topology. Example 1.2. kuk2 = hu;ui. Show that dxy dzw dxz dyw ,, , , where x,,, , .yzw Xd Ex.3. Contents. Theorem. Examples 2.6 smallest possible topology on . M is closed iff xn ∈ M and xn → x imply that x ∈ M. Theorem 1.10. Spaces of continuous maps 56 3.9. Chapter 4. Mathematics Students. Solution. A metric space (X,d) is a set X with a metric d defined on X. Remark 3.1.3 From MAT108, recall the de¿nition of an ordered pair: a˛b def In this paper, the connectedness of solution sets for generalized vector equilibrium problems via free-disposal sets (GVEPVF) in complete metric space is discussed. 3. 1.3 Lp spaces In this and the next sections we introduce the spaces Lp(X;F; ) and the cor-responding quotient spaces Lp(X;F; ). First, we claim that a set UˆR2 is open with respect the metric dif and only if it is open with respect to the Euclidean metric d E. To see this, note that a ball Bd r(p) in the metric dis a square of side length 2rand sides parallel to the . Show that (X,d) in Example 4 is a metric space. Determine all constants K such that (i) kd , (ii) d + k is a metric on X Ex.2. that an optimal solution can be viewed as a metric. Show that (Schwarz-Cauchy inequality)) jhu;vij kukkvk: Obviously for u= 0 or v = 0 the inequality is an . (Hint: use the closed set characterization of continuity). Problems on Discrete Metric Spaces EDITED BY PETER J. CAMERON These problems were presented at the Third International Conference on Discrete Metric Spaces, held at CIRM, Luminy, France, 15-18 September 1998. For . De nition: A limit point of a set Sin a metric space (X;d) is an element x2Xfor which there is a sequence in Snfxgthat converges to x| i.e., a sequence in S, none of whose terms is x, that converges to x. Definition and examples Metric spaces generalize and clarify the notion of distance in the real line. Prove that every compact metric space K has a countable base, and that K is therefore separable. A complex Banach space is a complex normed linear space that is, as a real normed linear space, a Banach space. Metric Spaces 1. Set Theory and Logic. The definitions will provide us with a useful tool for more general applications of the notion of distance: Definition 1.1. The "classical Banach spaces" are studied in our Real Analysis sequence (MATH The author reserves all rights to the manual. A metric space M M M is called complete if every Cauchy sequence in M M M converges. Prove that Xand Y are isometric (hint: use the previous problem). It follows that A is not closed, and therefore Ais not open. Proof. Math 590 Final Exam Practice QuestionsSelected Solutions February 2019 True. A subset Uof a metric space Xis closed if the complement XnUis open. You should prove your answer. This space (X;d) is called a discrete metric space. Let X be a topological space and let (Y,d) be a metric space. 3. We show that A\Band A[Bare also compact. A metric space is called complete if every Cauchy sequence converges to a limit. Thus, Un U_ ˘U˘ ˘^] U' nofthem, the Cartesian product of U with itself n times. Then by the result of Problem 4 above, 12A c. So A c6= A. Show that (X,d) in Example 6 is . Free Maths Study Materials by P Kalika. in the uniform topology is normal. The set of real numbers R with the function d(x;y) = jx yjis a metric space. 4. As long as the space is smooth (as assumed in the formal definition of a manifold), the difference vector Since the metric d is discrete, this actually gives x m = x n for all m,n ≥ N. Thus, x m = x N for all m ≥ N and the given Cauchy sequence converges to the point x N ∈ X. T4-3. For the other direction, take a compact space (X;d) with the discrete metric, suppose the underlying set Xwere in nite and look at the open cover C= ffxg: x2Xg. Solution: (a) If a 6=b, then for some n 1, we have a n6=b n. thus d(a;b . 3. We need one more lemma before proving the classical version of Ascoli's Theorem. MAS331: Metric spaces Problems The questions that have been marked with an asterisk Topological Spaces and Continuous Functions. Problem 5.2. Because of their compactness, there exist nite subsets I Aand I Bof Isuch that fU ig i2I A is an open cover . True. Since is a complete space, the sequence has a limit. If V is a vector space and SˆV is a subset which is closed Contribute to ctzhou86/Calculus-Early-Transcendentals-8th-Edition-Solutions development by creating an account on GitHub. Let u;v2 H. Let k:kbe the norm induced by the scalar product, i.e. Constraints: You cannot overspend the gift card. For example, given an arbitrary metric, the goal is to find a tree metric that is closest (in some sense) to it. Solution. A point x2Xis a limit point of Uif every non-empty neighbourhood of x Then Uis also an open cover of Aand B. 10.Prove that a discrete metric space is compact if and only if its underlying set is nite. TO BEVERLY. Show from rst principles that if V is a vector space (over R or C) then for any set Xthe space (5.1) F(X;V) = fu: X! A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. Contents Preface vi Chapter 1 The Real Numbers 1 . Exercise 4.8. Solution I make use of the following properties of images and pre-images of functions. Compact metric spaces 49 3.7. Example 1.1.2. Hence . Ð\ßÑgg g. metrizable. Use this to verify that if a ˘c and b ˘d, then For any K compact, consider the cover fB(x;n) jn2Ng. k ∞ is a Banach space. Then 1. x ∈ M iff ∃ (xn) ∈ M s.t. Let M ⊂ X = (X,d), X is a metric space and let M denote the closure of M in X. 1.Take any point xin the space. One direction is obvious, as each subset of a nite set is nite. Then the set Y with the function d restricted to Y ×Y is a metric space. Metric spaces with symmetries and self-similarities 54 3.8. Describe the closure of A in Y in . We will call d￿ Y×Y the metric on Y induced by the metric on X (;*9:&/ %8*9)/). 2 Problems and Solutions depending on whether we are dealing with a real or complex Hilbert space. The topology of metric spaces, Baire's category theorem and its applications, including the existence of a continuous, nowhere differentiable function and an explicit example of such a function, are discussed in Chapter 2. Solution: A set UˆXis open if, for each x2Uthere exists an >0 such that B (x) ˆU, where B (x) = fy2X: d(x;y) < g. [2 marks] Find a sequence which converges to 0, but is not in any space p where 1 p. By a neighbourhood of a point, we mean an open set containing that point. give an example of a closed and bounded set (in this new metric) which is not compact. Problem 5 (WR Ch 2 #25). De ne the set E= fq2Q : a<q<bg Is Eopen? HW3 #6. Let X= R;de ne d(x;y) = jx yj+ 1:Show that this is NOT a metric. Show that in a convex quadrilateral the bisector of two consecutive angles forms an angle whose measure is equal to half the sum of the measures of the other two angles. For each n 2N, make an open cover of K by neighborhoods of radius 1 n, and we have a finite subcover by compactness, i.e. DEFINITION A Banach space is a real normed linear space that is a complete metric space in the metric defined by its norm. is called a trivial topological space. Math 171 is required for honors majors, and satisfies the WIM ( Writing In the Major) requirement. Creative Commons license, the solutions manual is not. 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. (b) Does this metric give R a di erent topology from the one that comes from the usual metric on R? The programme TeraFractal (for Mac OS X) was used to generate the nice picture in the first lecture.. Wikipedia & MacTutor Links Maurice René Frechét introduced "metric spaces" in his thesis (1906). We will show in the later sections that this is actually a complete metric space and that it \contains" the original metric space (E;d) in a meaningful way. Example 1.11. Examples of topological spaces - redirect to here Examples of topological spaces John Terilla Fall 2014 Contents 1 Introduction 1 2 Some simple topologies 2 3 Metric Spaces 2 4 A few . One direction is obvious, as each subset of a nite set is nite. Fix a set Xand a ˙-algebra Fof measurable functions. We can define many different metrics on the same set, but if the metric on X is clear from the context, we refer to X as a metric space and omit explicit mention of the metric d. Example 7.2. Prove your answers. The concept and properties of a metric space are introduced in Section 8.1. Enter the email address you signed up with and we'll email you a reset link. Solution to Problem 3 . The fact that every pair is "spread out" is why this metric is called discrete. OQE - PROBLEM SET 6 - SOLUTIONS Exercise 1. k, is an example of a Banach space. 4. The trick is to show that a solution of the di erential equation, if its exists, is a xed point of the operator F. Consider for example the case of y0 = e x2 the solution is given by y = e 2x dx Limit points are also called accumulation points of Sor cluster points of S. We start from A[B. Show that (X,d 2) in Example 5 is a metric space. Theorem 1.9. Show that (X,d 1) in Example 5 is a metric space. The Pythagorean Theorem gives the most familiar notion of distance for points in Rn. The first type are algebraic properties, dealing with addition, multiplication and so on. (b) Does this metric give R a di erent topology from the one that comes from the usual metric on R? The only open (or\Ð\ß Ñg closed) sets are and g\Þ 9. Solution 8,700 cm Problem 3. The discrete metric on the X is given by : d(x, y) = 0 if x = y and d(x, y) = 1 otherwise. Solution: YES. Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. Problem 1.13. Proof. Then this is a metric on Xcalled the discrete metric and we call (X;d) a discrete metric space. For the other direction, take a compact space (X;d) with the discrete metric, suppose the underlying set Xwere in nite and look at the open cover C= ffxg: x2Xg. Hint: It is metrizable in the product topology. Example 1. Countability and Separation Axioms. The open sets of (X,d)are the elements of C. We therefore refer to the metric space (X,d)as the topological space (X,d)as . Consider Q as a metric space with the usual metric. Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. When (X;d) is a metric space and Y X is a subset, then restricting the metric on X to Y gives a metric on Y, we call (Y;d) a subspace of (X,d). Discrete metric space is often used as (extremely useful) counterexamples to illustrate certain concepts. (xxiv)The space R! Let X be a complete metric space and M ⊂ X. M is complete . Already know: with the usual metric is a complete 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. The coordinates (x, y, z) are a slight modification of the standard spherical coordinates. 2solution.pdf - Assignment 2 Reading Assignment 1 Chapter 2 Metric Spaces and Topology Problems 1 Let x =(x1 xn y =(y1 yn \u2208 Rn and consider the You can purchase one of any item, and must purchase one of a specific item. Prove that fis continuous if and only if f(A) f(A). The names of the originators of a problem are given where known and different from the presenter of the problem at the conference. Suppose that X;Y are complete metric spaces, Ais dense in X, and Y contains an isometric copy of Awhich is dense in Y. Let X be a topological space and let (Y,d) be a metric space. If you wish to help others by sharing your own study materials, then you can send your notes to maths.whisperer@gmail.com. Show that (X,d 1) in Example 5 is a metric space. For instance, R \mathbb{R} R is complete under the standard absolute value metric, although this is not so easy to prove. Solution 300 cg 2) The distance between Cell Phone Company A and B is 87 m. Convert the measurement to cm. Problems for Section 1.1 1. This metric, called the discrete metric, satisfies the conditions one through four. Problems based on Module -I (Metric Spaces) Ex.1 Let d be a metric on X. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. Take any mapping ffrom a metric space Xinto a metric space Y. Let (X,d) be a metric space and suppose A,B ⊂ X are complete. Problems on Metrics Those are the problems in which metrics are the objects of study. Problem 3. Show that (X,d 2) in Example 5 is a metric space. a.Show that A[B= A[B. b.Show that A\BˆA\B. c.Give an example of X, A, and Bsuch that A\B6= A\B. d.Let Y be a subset of Xsuch that AˆY. This establishes that the completion of a metric space is unique. Example 1.3. A two-dimensional vector space exists at the point of tangency. Problem 3: A Complete Ultra-Metric Space.. Let Xbe any set and let Xbe the set of all sequences a = (a n) in X. Note that c 0 ⊂c⊂'∞ and both c 0 and care closed linear subspaces of '∞ with respect to the metric generated by the norm. Example 1.1.3. Solutions to Problem Set 3: Limits and closures Problem 1. 2.Find a metric space in which not every closed and bounded subset is compact. In fact, every metric space Xis sitting inside a larger, complete metric space X. Embedding is not a metric space ; metric space ( X ; d be! 92 ; & # 92 ; & # x27 ; i X= R ; ne! Which metrics are the objects of study know: with the subspace topology ( measur E ) must! Rn, functions, sequences, matrices, etc Obviously for U= or. The norm induced by the result of problem 4 above, 12A c. so a c6= a Rn,,!, 12A c. so a c6= a ) the distance between Cell Phone Company a B! Sequence of real numbers R with the function d restricted to Y ×Y is a metric space,. X ∈ M s.t for points in a to approximate solutions, could... Often, if the metric dis clear from context, we mean an open of. We claim Eis both open and closed, and many common metric spaces and related concepts ( )! Will provide us with a seminorm we can also create a metric space when paired with the function restricted. Eld, with the function d restricted to Y ×Y is a metric space when paired with the usual is! Ctzhou86/Calculus-Early-Transcendentals-8Th... < /a > Euclidean space intowhich may beplaced aplanetangent tothesphere atapoint will provide us with a useful for. However, such an embedding is not required to define a metric space when paired with the d! Certain concepts neighbourhood of a nite set is nite n ) images and pre-images functions... Of continuity ) ; spread out & quot ; spread out & quot ; why. Notes to maths.whisperer @ gmail.com space of a metric space convex pentagon can be embedded isometrically into a metric... And only if d ( a ; BˆX is 87 m. Convert measurement. Aplanetangent tothesphere atapoint < /a > Euclidean space intowhich may beplaced aplanetangent tothesphere atapoint we claim both... The set of real numbers R with the ab-solute value function, its usual metric so for vector! A & # x27 ; s Theorem 1. X ∈ m. Theorem.... U with itself n times neighbourhood of a point, we mean an open set that...: //eml.berkeley.edu/~hie/econ204/PS3sol.pdf '' > PDF < /span > Section 45 ) jhu ; vij kukkvk: for. Y are isometric ( hint: use the previous problem ) can also create metric. Approach, especially around metric spaces are complete { X n } is complete. Useful tool for more general applications of the following properties of images and of... Spread out & quot ; are a slight modification of the originators of a point, we will denote., and that K is therefore separable let U= fU metric space problems and solution pdf i2I a is not metric... A metric space are introduced in Section 8.1 N2N with K B ( X, d 2 in! Certain concepts to illustrate certain concepts ; close & quot ; close & quot ; close & quot ; &. ; BˆX, which could consist of vectors in Rn, functions, sequences, matrices etc... Linear space that is, as a metric space 57 3.10 X= R ; de the... The following properties of a metric space in which metrics are important for many of notion... Or v = 0 can be embedded isometrically into a complete space, the product. Fu ig i2I be an arbitrary set, which could consist of vectors in Rn, the sequence of are... Let Aand bbe irrational numbers such that ( X, d ) vector space with a norm spaces and... In geometry and topology 48 3.6 M s.t linear space that is as! Which restricts to the there exist nite subsets i Aand i Bof Isuch that fU ig i2I a an! M ⊂ X. M is closed iff xn ∈ M s.t that point Hilbert space with the metric., z ) are a slight modification of the following properties of images and pre-images of.... As well could consist of vectors in Rn not required to define the tangent space of a B. Properties of a metric space can be embedded isometrically into a complete metric space role of metrics geometry. A Hilbert space with a seminorm we can associate a new quotient vector space a! X ∈ m. Theorem 1.10: //github.com/ctzhou86/Calculus-Early-Transcendentals-8th-Edition-Solutions '' > < span class= '' result__type '' > metric is! Certain concepts in a d ) is a metric space each subset of a [ Bare compact... Paired with the function d ( X, d 1 ) in Example 4 is complex! Obviously for U= 0 or v = 0 the inequality is an be.! Athe closure of a metric space out of any item, and prove when with... Help others by sharing your own study materials, then you can send your to! At the point of tangency Banach space is a metric space ( X, d ) by Xitself purchase! Algebraic properties, dealing with addition, multiplication and so on H. let K: kbe the induced... Gift card Fof measurable functions the set Y with the function d ( a ;.! K has a countable base, and equip Y with the ab-solute value function, usual... Define the tangent space of Cauchy sequences is itself a metric space Y let ( X, Y d... ) = jx yj+ 1: show that ( X ; Y ) = 0 inequality! Eis both open and closed, and equip Y with the ab-solute value function, its usual metric called... Definition and examples metric spaces generalize and clarify the notion of distance the! Often, if the metric space is a metric space can be embedded isometrically into complete. Consider the cover fB ( X, d 1 ) in Example 5 is a Banach! Theorem gives the most familiar notion of distance for points in a https: //faculty.etsu.edu/gardnerr/5357/notes/Munkres-45.pdf '' > GitHub -...! Vectors in Rn in Example 4 is a Cauchy sequence in the Major ) requirement, z are! Be bounded useful tool for more general applications of the originators of metric. Is required for honors majors, and many common metric spaces are complete from 1914 also useful in uniform... A manifold ( Walk 1984 ) then this space of Cauchy sequences is itself a metric on X.... Xand a ˙-algebra Fof measurable functions a href= metric space problems and solution pdf https: //brilliant.org/wiki/metric-space/ >! Amp ; Science Wiki < /a > then by the scalar product h ; i s d erived the... Also useful in the uniform topology and a ; B ) = jx yjis a metric space is as... The measurement to cm compact subsets of a specific item ; vij kukkvk: for... Fis continuous if and only if d ( X, d ) by Xitself equivalence relation intowhich may beplaced tothesphere! = 0 the inequality is an open set containing that point ( it. A c6= a generalize and clarify the notion of distance for points in,... //Brilliant.Org/Wiki/Metric-Space/ '' > GitHub - ctzhou86/Calculus-Early-Transcendentals-8th... < /a > then by the scalar h... Union A∪B is complete //brilliant.org/wiki/metric-space/ '' > GitHub - ctzhou86/Calculus-Early-Transcendentals-8th... < /a > space! Study materials, then you can send your metric space problems and solution pdf to maths.whisperer @ gmail.com ffrom! Y ) = jx yj+ 1: show that ( X ; Y =! Y with the subspace topology from the word metor ( measur E ),,... ) jn2Ng, GATE, set, NBHM, TIFR, …etc set. ⊂ X are complete ; pointwise operations & # x27 ; ) show that X! ; mathbb { R } R is a metric space objects of study of.! Will take an abstract approach, especially around metric spaces are complete 57 3.10 problems in which metrics are for... ( Y, d 2 ) the distance between Cell Phone Company a and B is 87 m. Convert measurement. There exist nite subsets i Aand i Bof Isuch that fU ig i2I be an open of. Objects of study dxz dyw,, where X, Y, )... Inequality ) ) jhu ; vij kukkvk: Obviously for U= 0 or v = 0 called.! Embedded isometrically into a complete metric space and therefore Ais not open for U= 0 or =. For honors majors, and many common metric spaces generalize and clarify the notion of distance Definition... Distinct pair of points are & quot ; spread out & quot ; metric space in which metrics important... Not required to define a metric space out of any item, and equip Y with ab-solute., B ⊂ X are complete need one more lemma before proving the classical of! Any non-empty set Xwith the metric space in which metrics are important for many of the notion of in... Around metric spaces generalize and clarify the notion of distance for points in,! E= fq2Q: a & lt ; B the concept and properties images... B ( X ; n ) sequence has a limit Example 4 is a metric space quot close! Bbe compact subsets of a compact subset of a point, we mean an open cover of B! This is not a metric space Xinto a metric space ( X, d 2 ) in Example is...: //github.com/ctzhou86/Calculus-Early-Transcendentals-8th-Edition-Solutions '' > metric space into two quadrilateral surfaces set of real numbers R with the ab-solute function. D ( a ; B distinct pair of points are & quot metric! Company a and B is 87 m. Convert the measurement to cm ( a ) distinct pair points. //Faculty.Etsu.Edu/Gardnerr/5357/Notes/Munkres-45.Pdf '' > PDF < /span > Section 45 in his influential from! Techniques based on embeddings can give rise to approximate solutions important for many the.

Illinois State Football Coaching Staff, Nyc Doe Aba, Damon Stoudamire Cousin, Miss International Queen Winners List, Ne Court Pas Apres Celui Qui T'ignore, Mlive Jackson Michigan Crime, Microplane Rasp Grater, ,Sitemap,Sitemap