 # metric spaces homework solutions

1 ) 8 " > 0 9 N 2 N s.t. x 1 (n ! The “largest” and the ‘smallest” are in the sense of inclusion ⊂. What could we say about the properties of the metric spaces i described above in the spirit of the description of the continuity of the real line? Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. Question: (a) State The Stone-Weierstrass Theorem For Metric Spaces. math; advanced math; advanced math questions and answers (a) State The Stone-Weierstrass Theorem For Metric Spaces. Solution. The resulting measure is the unnormalized s-Hausdorff measure. 4.1.3, Ex. Problem 4.10: Use the fact that infinite subsets of compact sets have limit points to give an alternate proof that if X and Z are metric spaces with X compact, and f: X → Z is continuous, then f is uniformly continuous. 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. For n2P, let B n(0) be the ball of radius nabout 0 with respect to the relevant metric on X. For Euclidean spaces, using the L 2 norm gives rise to the Euclidean metric in the product space; however, any other choice of p will lead to a topologically equivalent metric space. The case of Riemannian manifolds. Recall that we proved the analogous statements with ‘complete’ replaced by ‘sequentially compact’ (Theorem 9.2 and Theorem 8.1, respectively). MA 472 G: Solutions to Homework Problems Homework 9 Problem 1: Ultra-Metric Spaces. Hint: It is metrizable in the uniform topology. In this case, we say that x 0 is the limit of the sequence and write x n := x 0 . Let Mbe a compact metric space and let fx ngbe a Cauchy sequence in M. By Theorem 43.5, there exists a convergent subsequence fx n k g. Let x= lim k!1 x n k. Since fx ngis Cauchy, there exists some Nsuch that m;n Nimplies d(x m;x n) < 2. De¿nition 5.1.1 Suppose that f is a real-valued function of a real variable, p + U, and there is an interval I containing p which, except possibly for p is in the domain of f . Let (X,d) be a metric space, and let C(X) be the set of all continuous func-tions from X into R. Show that the weak topology deﬁned on X by the functions in C(X) is the given topology on X deﬁned by the metric. Solution. Show that g fis continuous at p. Solution: Let >0 be given. [0;1] de ned by f a(t) = (1 if t= a 0 if t6=a There are uncountably many such f a as [0;1] is uncountable. d(x n;x 1) " 8 n N . A function d: X X! True. Note: When you solve a problem about compactness, before writing the word subcover you need to specify the cover from which this subcover is coming from 58. True. R is an ultra-metric if it satis es: (a) d(a;b) 0 and d(a;b) = 0 if and only a= b. 5.1.1 and Theorem 5.1.31. Metric Spaces MT332P Problems/Homework/Notes Recommended Reading: 1.Manfred Einsiedler, Thomas Ward, Functional Analysis, Spectral Theory, and Applications In mathematics, a metric space is a set together with a metric on the set. 5.1 Limits of Functions Recall the de¿nitions of limit and continuity of real-valued functions of a real vari-able. Give an example of a bounded linear operator that satis es the Fredholm alternative. Find solutions for your homework or get textbooks Search. Consider R with the usual topology. Let Xbe a set. Let 0 = (0;:::;0) in the case X= Rn and let 0 = (0;0;:::) in the case X= l1; l2; c 0;or l1. 5. (xxiv)The space R! Let X= Rn;l1;l2;c 0;or l1. I am not talking about the definition which is an abstraction, i am talking about the application of the definition like above in the real line. mapping metric spaces to metric spaces relates to properties of subsets of the metric spaces. See, for example, Def. I will post solutions to the … Solution: Only the triangle inequality is not obvious. (b) d(a;b) = d(b;a). Let (X,d) be a metric space. Problem 14. Solutions to Assignment-3 September 19, 2017 1.Let (X;d) be a metric space, and let Y ˆXbe a metric subspace with the induced metric d Y. Does this contradict the Cantor Intersection Theorem? Let us write D for the metric topology on … Solution: (a) Assume that there is a subset B of A such that B is open, A ⊂ B, and A 6= B. If (x n) is Cauchy and has a convergent subsequence, say, x n k!x, show that (x n) is convergent with the limit x. Hint: Homework 14 Problem 1. Let EˆY. Banach spaces and Hilbert spaces, bounded linear operators, orthogonal sets and Fourier series, the Riesz representation theorem. Proof. It remains to show that D satisﬁes the triangle inequality, D(x,z) ≤ D(x,y)+D(y,z). Whatever you throw at us, we can handle it. This is to tell the reader the sentence makes mathematical sense in any topo-logical space and if the reader wishes, he may assume that the space is a metric space. Solution. SECTION 7.4 COMPLETE METRIC SPACES 31 7.4 Complete Metric Spaces I Exercise 64 (9.40). For instance, R \mathbb{R} R is complete under the standard absolute value metric, although this is not so easy to prove. Let Xbe a metric space and Y a subset of X. Similar to the proof in 1(a) using the fact that ! (a) Prove that if Xis complete and Yis closed in X, then Yis complete. Answers and Replies Related Topology and Analysis News on Phys.org. Show that: (a) A is the largest open set contained in A. Let (M;d) be a complete metric space (for example a Hilbert space) and let f: M!Mbe a mapping such that d(f(m)(x);f(m)(y)) kd(x;y); 8x;y2M for some m 1, where 0 k<1 is a constant. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. Homework 2 Solutions - Math 321,Spring 2015 (1)For each a2[0;1] consider f a 2B[0;1] i.e. Let X D.0;1“. Convergent sequences are defined (in arbitrary topological spaces in Munkres 2.17, specifically on page 98 - to get the definition of metric space, replace "for each open U" by "for each epsilon ball B(x,epsilon)" in the definition.). in the uniform topology is normal. Let F n.0;1=n“for all n2N. Let X, Y, and Zbe metric spaces, with metrics d X, d Y, and d Z. A metric space is a set X together with a function d (called a metric or "distance function") which assigns a real number d(x, y) to every pair x, y X satisfying the properties (or axioms): d(x, y) 0 and d(x, y) = 0 x = y, d(x, y) = d(y, x), d(x, y) + d(y, z) d(x, z). Let (X,d) be a metric space and let A ⊂ X. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. (b) A is the smallest closed set containing A. Give an open cover of B1 (0) with no finite subcover 59. 130 CHAPTER 8. Spectrum of a bounded linear operator and the Fredholm alternative. In a complete metric space M, let d(x;y) denote the distance. A metric space M M M is called complete if every Cauchy sequence in M M M converges. Math 104 Homework 3 Solutions 9/13/2017 3.We use the Cauchy{Schwarz inequality with b 1 = b 2 = = b n= 1: ja 1 1 + a 2 1 + + a n 1j q a2 1 + a2 2 + + a2 p n: On the other hand, ja 1 1 + a 2 1 + + a n1j= ja 1 + a 2 + + a nj 1: Combining these two inequalities we have 1 q a 2 1 + a 2 + + a2n p Let f: X !Y be continuous at a point p2X, and let g: Y !Z be continuous at f(p). Assume there is a constant 0 < c < 1 so that the sequence xk satis es d(xn+1; xn) < cd(xn; xn 1) for all n = 1;2;:::: a) Show that d(xn+1;xn) < cnd(x1;x0). Solution. (c)For every a;b;c2X, d(a;c) maxfd(a;b);d(b;c)g. Prove that an ultra-metric don Xis a metric on X. Take a point x ∈ B \ A . The metric space X is said to be compact if every open covering has a ﬁnite subcovering.1 This abstracts the Heine–Borel property; indeed, the Heine–Borel theorem states that closed bounded subsets of the real line are compact. solution if and only if y?ufor every solution uof Au= 0. Metric spaces and Multivariate Calculus Problem Solution. (xxv)Every metric space can be embedded isometrically into a complete metric space. Solution. Solutions to Homework 2 1. Then fF ng1 nD1 is a descending countable collection of closed, … Defn A sequence {x n} in a metric space (X,d) is said to converge, to a point x 0 say, if for each neighborhood of x 0 there exists a natural number N so that x n belongs to the neighborhood if n is greater or equal to N; that is, eventually the sequence is contained in the neighborhood. Solution: It is clear that D(x,y) ≥ 0, D(x,y) = 0 if and only if x = y, and D(x,y) = D(y,x). Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) Compactness in Metric Spaces: Homework 5 atarts here and it is due the following session after we start "Completeness. MATH 4010 (2015-16) Functional Analysis CUHK Suggested Solution to Homework 1 Yu Meiy P32, 2. Home. Differential Equations Homework Help. Is it a metric space and multivariate calculus? True. Show that the functions D(x,y) = d(x,y) 1+d(x,y) is also a metrics on X. The following topics are taught with an emphasis on their applicability: Metric and normed spaces, types of convergence, upper and lower bounds, completion of a metric space. Prove that none of the spaces Rn; l1;l2; c 0;or l1is compact. View Test Prep - Midterm Review Solutions: Metric Spaces & Topology from MTH 430 at Oregon State University. (xxvi)Euclidean space Rnis a Baire space. Homework Statement Is empty set a metric space? 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. Homework Equations None. Let X be a metric space and C(X) the collection of all continuous real-valued functions in X. 0. The metric satisfies a few simple properties. Knowing whether or not a metric space is complete is very useful, and many common metric spaces are complete. Homework 7 Solutions Math 171, Spring 2010 Henry Adams 42.1. Prove that a compact metric space is complete. SOLUTIONS to HOMEWORK 4 Problem 1. f a: [0;1] ! It covers the topology of metric spaces, continuity, connectedness, compactness and product spaces, and includes results such as the Tietze-Urysohn extension theorem, Picard's theorem on ordinary differential equations, and the set of discontinuities of the pointwise limit of a sequence of continuous functions. Homework 3 Solutions 1) A metric on a set X is a function d : X X R such that For all x, As an example, consider X= R, Y = [0;1]. Here are instructions on how to submit the homework and take the quizzes: Homework + Quiz Instructions (Typo: Quizzes are 8:30-8:50 am PST) Note: You can find hints and solutions to the book problems in the back of the book. 4.4.12, Def. Let ( M;d ) be a metric space and ( x n)n 2 N 2 M N. Then we de ne (i) x n! In the category of metric spaces (with Lipschitz maps having Lipschitz constant 1), the product (in the category theory sense) uses the sup metric. Solutions to Homework #7 1. Thank you. 46.7. Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. Our arsenal is the leading maths homework help experts who have handled such assignments before and taught at various universities around the UK, the USA, and Canada on the same topic. Since x= lim k!1 x n k, there exists some Kwith n EUCLIDEAN SPACE AND METRIC SPACES 8.2.2 Limits and Closed Sets De nitions 8.2.6. (b) Prove that if Y is complete, then Y is closed in X. SOLUTIONS to HOMEWORK 2 Problem 1. A “solution (sketch)” is too sketchy to be considered a complete solution if turned in; varying amounts of detail would need to be filled in. The Attempt at a Solution It seems so because all the metric properties are vacuously satisfied. 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Y a subset of X Solutions: metric spaces 8.2.2 Limits and closed De. Embedded isometrically into a complete metric space is a set together with a metric and!