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Homework Help: Topological isomorphism

  1. Oct 4, 2008 #1
    1. The problem statement, all variables and given/known data

    http://img219.imageshack.us/img219/2512/60637341vi6.png [Broken]



    2. Relevant equations
    I think this is relevant:
    http://img505.imageshack.us/img505/336/51636887dc4.png [Broken]



    3. The attempt at a solution
    A topological isomorphism implies that T and T-1 are bounded and given is that all cauchy sequences in E are convergent.

    But if that's the case then by boundedness all convergent sequences are bounded in het norm of F and are thus also Cauchy seqeunces. Am I thinking in the right direction?
     
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Oct 4, 2008 #2

    morphism

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    You have to prove that every Cauchy sequence in F converges. How is what you're saying doing that?
     
  4. Oct 4, 2008 #3
    Okay so that's wrong but suppose I have a cauchy sequence in E: [tex]|x_n-x_m|_E < \epsilon\ , \forall n,m\geq N [/tex] how can I prove then that F is also Banach?
     
  5. Oct 4, 2008 #4

    morphism

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    Why are you taking a cauchy sequence in E? You're supposed to prove that F is a Banach space.
     
  6. Oct 5, 2008 #5
    Because it is given that E is Banach what implies that every cauchy sequence converges.
     
  7. Oct 5, 2008 #6

    morphism

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    What I meant is that it makes more sense to start with a cauchy sequence in F rather than in E. Because you want to prove that every cauchy sequence in F converges.
     
  8. Oct 6, 2008 #7
    Can someone give me a hint? Because I've started with a cauchy sequence in F but I honestly do not see what to do next.
     
  9. Oct 6, 2008 #8

    HallsofIvy

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    Let {an} be a sequence in F. What can you say about {T-1 an}?
     
  10. Oct 7, 2008 #9

    Let [tex] \{ a_n \}[/tex] be a sequence in F then for all [tex] n,m \geq N [/tex] we have:


    [tex] || T^{-1} (a_n-a_m)||_E \leq c\cdot ||a_n-a_m||_F < c \cdot \epsilon [/tex]

    So an is Cauchy in F.

    But how do I get it to converge in F with limit a?
     
  11. Oct 7, 2008 #10

    morphism

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    Look, you want to show that F is a Banach space. So you take an arbitrary cauchy sequence in F and show that it converges in F. What do we have to work with here? We know that F is isomorphic to a Banach space. Use that isomorphism.
     
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