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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.

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

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