Question

then e is 0U tric space. Show that there is an isometric imbedding h of X 、D), as follows: Let X denote the set of all space

#9. all one problem.

0 0
Add a comment Improve this question Transcribed image text
Answer #1

be he Given that (x,d) is a metic Spate et of all Cauchy Sequentes Define īfdc.end.) → o asn--) oo х~Y all e2hi valene. ClayNoo consder Well-ARne』 is D then T93 Ts.J and )山(iii) D(r,[i]6 his one-one 게 so h is one- on e (a) h is an isomelσ h(x) is Jense in y Senente Considod Thor is caud So hex) is ense in avc

Add a comment
Know the answer?
Add Answer to:
#9. all one problem. then e is 0U tric space. Show that there is an isometric...
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for? Ask your own homework help question. Our experts will answer your question WITHIN MINUTES for Free.
Similar Homework Help Questions
  • 8. A subsetD of a metric space X is dense if for all E X and...

    8. A subsetD of a metric space X is dense if for all E X and all e E R+ there is an element yE D such that d(x, y) <. Show that if all Cauchy sequences (yn) from a dense set D converge in X, then X is complete.

  • Please answer c d e 3. This problem shows that the metric space of continuous real-valued...

    Please answer c d e 3. This problem shows that the metric space of continuous real-valued functions C([0, 1]) on the interval [0, 1is complete. Recall that we use the sup metric on C([0,1), so that df, 9) = sup{f (2) - 9(2): € (0,1]} (a) Suppose that {n} is a Cauchy sequence in C([0,1]). Show that for each a in 0,1], {Sn(a)} is a Cauchy sequence of real numbers. (b) Show that the sequence {fn(a)} converges. We define f(a)...

  • B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Def...

    B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...

  • (5) Here is a fascinating equivalence for being a complete metric space that we will use...

    (5) Here is a fascinating equivalence for being a complete metric space that we will use later. Let (X,d) be a metric space. (b) ** (10 points) Show that the following are equivalent: • (X, d) is complete; • for every family of non-empty closed subsets Fo, F1, F2, ... of X such that F, 2 F12 F22... and limn700 diam( Fn) = 0, it holds that Nnen Fn = {a} for some a € X. (Hint: for the reverse...

  • problem 23 please :) and here is Q.21 Problem 23. Recall from Problem 21 the equivalence...

    problem 23 please :) and here is Q.21 Problem 23. Recall from Problem 21 the equivalence relation ~ on the set of rational Cauchy sequences C. Define 〈z) E C to be eventually positive if there is an M є N such that xn > 0 for all Prove that eventually positive is a well defined notion on c/ (z〉 ~ 〈y), then 〈y〉 İs eventually positive. ie. if 〈z) is eventually positive and Problem 21. Let C be the...

  • 2. Let M be the set of all measurable sets in R, and let d be our semi-metric, show that (M, d) i...

    please explain the steps you take 2. Let M be the set of all measurable sets in R, and let d be our semi-metric, show that (M, d) is complete: If (An)1 is a Cauchy sequence (with our semi- metric d) then there is a measurable set A EM such that lim, too d(An, A) 0. 2. Let M be the set of all measurable sets in R, and let d be our semi-metric, show that (M, d) is complete:...

  • Problem 1. Let (X, d) be a metric space and t the metric topology on X....

    Problem 1. Let (X, d) be a metric space and t the metric topology on X. (a) Fix a E X. Prove that the map f :(X, T) + R defined by f(x) = d(a, x) is continuous. (b) If {x'n} and {yn} are Cauchy sequences, prove that {d(In, Yn)} is a Cauchy sequence in R.

  • Exercise 5 (based on Tao). Let (X,d) be an arbitrary metric space. Prove the following statements...

    Exercise 5 (based on Tao). Let (X,d) be an arbitrary metric space. Prove the following statements (1) If a sequence is convergent in X, all its subsequences are converging to the same limit as the original sequence. (2) If a subsequence of a Cauchy sequence is convergent, then the whole sequence is convergent to the same limit as the subsequence. (3) Suppose that (X,d) is complete and Y S X is closed in (X,d). Then the space (Y,dlyxy) is complete....

  • Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d...

    Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.) Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...

  • A subset D of a metric space (X, d) is dense if every member of X...

    A subset D of a metric space (X, d) is dense if every member of X is a limit of a sequence of elements from D. Suppose (X,d) and (Y,ρ) are metric spaces and D is a dense subset of X. 1. Prove that if f : D -» Y is uniformly continuous then there exists an extension15 of f to a if dn (E D) e X define 7(x) lim f(d,) uniformly continuous function f:X * Y. Hint: 2....

ADVERTISEMENT
Free Homework Help App
Download From Google Play
Scan Your Homework
to Get Instant Free Answers
Need Online Homework Help?
Ask a Question
Get Answers For Free
Most questions answered within 3 hours.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT