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?