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Continuity of one Norm w.resp. to Another. Meaning?

  1. Jul 18, 2012 #1

    Bacle2

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    Hi, All:

    I am working on a proof of the fact that any two norms on a f.dim. normed space V are equivalent. The idea seems clear, except for a statement that (paraphrase) any norm in V is a continuous function of any other norm. For the sake of context, the whole proof goes like this:

    1)Show that , if any two norms f,g on the unit sphere on V are equivalent, then f,g are equivalent in the whole of V. Easy; just rescaling.

    2)**Every norm is a function of some other norm **

    3) In particular, from 2, f,g are both cont. functions of another norm,say, h.

    4)We define the function j on the unit sphere S^1 of V by : j=f/g . Then, by compactness of


    S^1 , there are constants m, M with 0<m<=M with m<j<M , i.e., m<f/g < M

    Not too hard.

    Still, I'm having trouble pinning-down the meaning of the statement in 2. Any Ideas?

    Thanks.
     
  2. jcsd
  3. Jul 19, 2012 #2
    It just means that for every norm f on V, there exists a continuous function F:ℝ→ℝ and a norm h on V, such that [itex]f=F\circ h[/itex].
     
  4. Jul 20, 2012 #3

    Bacle2

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    But this seems like a very stringent condition. After some reflection with my friend

    double-espresso , I think it is more reasonable that , given a fixed norm N and other

    norms f,g as above, that we can say that f,g are continuous functions in the

    topology that N gives rise to , when we use the metric d(x,y)=N(x-y) , don't

    you think this seems more reasonable ?
     
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