1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

2 definitions and a theorem relating them

  1. Oct 21, 2012 #1

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    1. The problem statement, all variables and given/known data

    Let E be a vector space and p,q 2 norms on it. By definition p,q induce the same topology on E, iff they assign the same neighborhood basis to the 0 vector. *QUESTION: What does the bolded part mean ? Does this mean that, if whatever A included in the system (=basis?) induced by p, A is included in a certain B in the system induced by q and viceversa, i.e. any B in the system induced by q is included in a certain A induced by p ? If so, then see my arguments below.

    The 2nd definition would be:

    E a vector space and p, q norms on it. p, q are said to be equivalent iff there exist c, C>0 such as

    [tex] c p(x) \leq q(x) \leq C p(x), ~\forall x \in E [/tex]

    The theorem would be the equivalence of the 2 definitions, namely

    Theorem: p,q are equivalent iff they induce the same topology on E.

    The problem would be to prove the theorem.

    3. The attempt at a solution

    Assume p, q equivalent. Then if [itex]\epsilon >0 [/itex], [itex] \{ x\in E, p(x) \leq \epsilon/c \}[/itex] is a n-hood system of 0 wrt p. Let A be one of the n-hoods in this system. Then [itex] A\subseteq \{x\in E, q(x) \leq\epsilon \} [/itex] ?? That would mean that [itex] \{x\in E, q(x) \leq\epsilon \} [/itex] is also a n-hood system of 0 induced by q. Using now the 2nd inequality would mean

    Let [itex] \epsilon>0 [/itex]. Then [itex] \{x\in E, q(x)\leq \epsilon/C \} is a n-hood system of 0 induced by q and let B be a n-hood from it. Does it mean (using the 2nd inequality in the definition of equivalence) that:

    [tex] B\subseteq \{x\in E, p(x)\leq \epsilon\}[/tex] ??

    If so, then the 2 systems of n-hoods would be the same (??) so that p,q induce the same topology.

    This would be half the proof. How's the other half ?
     
    Last edited: Oct 21, 2012
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: 2 definitions and a theorem relating them
  1. 2 definitions (Replies: 1)

Loading...