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Homework Help: 2 definitions and a theorem relating them

  1. Oct 21, 2012 #1

    dextercioby

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    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
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