this only to have some intuition for how to think about metric spaces in general, but that anything we prove about metric spaces must be phrased solely in terms of the de nition of a metric itself. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … As we said, the standard example of a metric space is Rn, and R, R2, and R3 in particular. Can a function whose points are all local minima can be non-constant? A sequence (x n) in X is called a Cauchy sequence if for any ε > 0, there is an n ε ∈ N such that d(x m,x n) < ε for any m ≥ n ε, n ≥ n ε. Theorem 2. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. Proof. This theorem implies that the completion of a metric space is unique up to isomorphisms. Corollary 1.2. In fact, a metric space is compact if and only if it is complete and totally bounded. Let (X,d) be a metric space. Find out what you can do. Now we’ll prove that R is a complete metric space, and then use that fact to prove that the Euclidean space Rn is complete. Proof: Let fx ngbe a Cauchy sequence. Prove problem 2. \begin{align} \quad d(x_n, p) \leq d(x_n, x_m) + d(x_m, p) < \epsilon_1 + \epsilon_1 = \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align} Complete Metric Spaces Deﬁnition 1. Prove that R^n is a complete metric space. Let f= j 2 j 1 1: j 1(X) !X 2:Then fis an isometry. By Theorem 13, C b(X;Y) is a closed subspace of the complete metric space B(X;Y), so it is a complete metric space. 1) is a complete metric space. In order to prove that R is a complete metric space, we’ll make use of the following result: Proposition: Every sequence of real numbers has a monotone subsequence. A sequence (x n) in X is called a Cauchy sequence if for any ε > 0, there is an n ε ∈ N such that d(x m,x n) < ε for any m ≥ n ε, n ≥ n ε. Theorem 2. Example 2. Proof. Let ε > 0 be given. Given any isometry f: X!Y into a complete metric space Y and any completion (X;b d;jb ) of (X;d) there is a unique isometry F: Xb !Y such that f= F j: Proof. Prove that R^n is a complete metric space. Theorem: R is a complete metric space | i.e., every Cauchy sequence of real numbers converges. This is a generalization of the Heine–Borel theorem, which states that any closed and bounded subspace S of R n is compact and therefore complete. Proof. However, we can put other metrics on these sets beyond the standard ones. Prove problem 2. Now we’ll prove that R is a complete metric space, and then use that fact to prove that the Euclidean space Rn is complete. 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. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. A metric space X is said to be complete if every Cauchy sequence in X converges to an element of X. (Universal property of completion of a metric space) Let (X;d) be a metric space. Proof. Indeed, a space is complete if and only if it is closed in any containing metric space. Assume that (x n) is a sequence which converges to x. Remark 1 ensures that the sequence is bounded, and therefore that every subsequence is bounded. Proof: Let fx ngbe a Cauchy sequence. We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. Let (X, d) be a complete metric space. The sequence fx ngmust have a rst term, say x n 1, such that all subsequent terms are … Any convergent sequence in a metric space is a Cauchy sequence. … A metric space X is said to be complete if every Cauchy sequence in X converges to an element of X. 4 Continuous functions on compact sets De nition 20. Now let us go back to prove that any two completions of a metric space are isomorphic. Complete Metric Spaces Deﬁnition 1. Metric Spaces A metric space is a set X that has a notion of the distance d(x,y) between every pair of points x,y ∈ X. \begin{align} \quad d(x_n, p) \leq d(x_n, x_m) + d(x_m, p) < \epsilon_1 + \epsilon_1 = \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align} Any convergent sequence in a metric space is a Cauchy sequence. Since X 2 is complete, fcan be extended to an isometry F: X 1!X 2:Let us prove that Fis surjective. Let (X,d) be a metric space. Bounded and totally bounded spaces If you want to discuss contents of this page - this is the easiest way to do it. The purpose of this chapter is to introduce metric spaces and give some deﬁnitions and examples. Proof: Suppose the sequence fx nghas no monotone increasing subsequence; we show that then it must have a monotone decreasing subsequence. Theorem: R is a complete metric space | i.e., every Cauchy sequence of real numbers converges.

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