uniform norm:

i think this is obvious:

but i cant find a positive real number a,

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- Nov 1st 2008, 09:26 AMszpengchaois this norm lipschitz equivalent to uniform norm?

uniform norm:

i think this is obvious:

but i cant find a positive real number a, - Nov 1st 2008, 10:31 AMLaurent
This is because there exists no such : for every you can easily find such that and : sketch the graph of a piecewise linear function that quickly decreases to 0 and then is constant equal to 0. (Or choose for large enough if you prefer to write formulas (then ))

- Nov 1st 2008, 10:32 AMszpengchaook
ok. so they are not lipschitz equivalent then?

- Jul 13th 2010, 06:37 PMelmbut "a" can be arbitrarily small too...
i don't get it. "a" can be arbitrarily small...

can't you do the following... let a -> 0+ and let A-> positive infinity. then it must be true that (a * uniform norm) <= norm1 <= (A * uniform norm) since 0 <= norm1 <= infinity, for all f.

what's wrong with that?

i understand there's a theory that all norms in Rn are Lipschitz equivalent to the Euclidian norm. then all should be Lipschitz equivalent to each other. seems applicable?