Question: A metric space if compact if and only if it is complete and totally bounded. x=Us

A metric space if compact if and only if it is complete and totally bounded.
A metric space if compact if and only if it
A metric space if compact if and only if it

x=Us

Step by Step Solution

★★★★★

3.53 Rating (163 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer
Step: 1 Unlock

Assume is compact The previous exercise showed that is totally bounded Further since every seque... View full answer

blur-text-image
Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock
Step: 3 Unlock

Document Format (1 attachment)

Word file Icon

914-M-N-A-O (95).docx

120 KBs Word File

Students Have Also Explored These Related Numerical Analysis Questions!

Q:

Let X be a metric space. a) Prove that if E X is compact, then E is sequentially compact (see Exercise 10.1.10). b) Prove that if X is separable and satisfies the Bolzano-Weierstrass Property, then...

Q:

a) Prove that Cantor's Intersection Theorem holds for nested compact sets in an arbitrary metric space; that is, if H1, H2, ... is a nested sequence of nonempty compact sets in X, then b) Prove that...

Q:

Let X be a compact metric space. A closed subspace of C(X) is compact if and only if it is bounded and equicontinuous.

Q:

8. (a) Prove that Cantor's Intersection Theorem holds for nested compact sets in an arbitrary metric space; i.e., if Hl , H2 , ... is a nested sequence of nonempty compact sets in X, then 00 k=l (b)...

Q:

Problem 2. A metric space {X, d) is called sepamble if it contains a countable dense subset, i.e. there exists a countable set S C X with g = X. {a} Show that the real line with the usual {Euclidean}...

Q:

I am really desperate for someone to answer all 4 parts of this question (d, e, f, and g) (I have already posted this question twice and no one has answered all 4 parts sufficiently). This is due in...

Q:

Solve Supplementary Problems COMPLETE METRIC SPACES 22. Let (X, d) be a metric space and let e be the metric on X defined by e(a, b) = min {1, d(a, b)). Show that (a,) is a Cauchy sequence in (X, d)...

Q:

MAT 327 Homework assignment 3. Due Thursday, October 29, 2015 Each problem is worth 10 points Problem 1. Give an example of a metric space X and a closed and bounded subspace Y of X that is not...

Q:

. Answer the following questions. 1. (3 points) In Metropolis only taxicabs and privately owned automobiles are allowed to use the highway betwem the airport and downtown. The market for taxi cab...

Q:

Kindly solve the following questions. 8. Find a limit point for the set {-1 : ne ( ). Bolzano 9. Let A = {-1 : ne N}. a) Find A in R. b) Explain why A is a compact subset of H. 10. Let X be a compact...

Q:

Describe what happens in the norming/third stage of team development.

Q:

The Basic Values the Filipinos (Maka-kalikasan) Identify three (3) environmental issues or concerns in your community and determine how NSTP and student-trainees could address solutions to each....

Q:

Long - term stock investment strategies include a . buy and hold. b . dollar - cost averaging. c . dividend reinvestment. d . a . and b . e . a . , b . , and c .

Q:

The key inputs to the Validate Scope process include all the below items EXCEPT a. The project management plan (scope management plan and scope baseline) b. Change requests c. Verified deliverables...

Q:

If A = [a1, b1] x . x [an, bn] and f: A R is continuous, define f: A R by

Q:

Let f: [a, b] x [c, d] R be continuous and suppose D 2 f is continuous. Define f (y) = ba f (x, y) dx. Prove Leibnitz' Rule: f1 (y) = ba D2 f (x, y) dx.

Q:

If f: [a, b] x [c, d] R is continuous and D2f is continuous, define F (x, y) = xa (t,y) dt a. Find D1F and D2F. (b) If G (x) = g(x) f (t, x) dt, find G1 (x).

Q:

A food safety guideline is that the mercury in fish should be below 1 part per million (ppm) Listed below are the amounts of mercury (ppm) found in tuna sushi sampled at different stores in a major...

Q:

Question 33 Test the claim that the proportion of men who own cats is significantly different than the proportion of women who own cats at the 0.02 significance level. The null and alternative...

Q:

help please thanks Consider the relationship between the number of bids an item on ebay received and the item's selling price. The following is a sample of 3 items sold through an auction. Price in...