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This is with reference to theorem 2.18 that appears here http://www.maths.gla.ac.uk/~ajb/dvi-ps/padicnotes.pdf

Essentially the author says that if $N$ is a norm on a ring $R$ and $\hat{R}$ is the completion of $R$ with respect to this norm, then there is a unique way to extend the norm to $\hat{R}$. I don't quite understand the uniqueness part and I don't think the author actually proves this.

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It seems the uniqueness is essentially implied in the statement since R is dense in R^. –  Fredrik Meyer May 5 '11 at 1:04

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