1. Show that two norms are equivalent iff the convergence of a sequence in the first norm is equivalent to its convergence in the second norm.
2. Let be the space of all real valued functions on which have continuous derivatives and define the norm by:
Is it a Banach spae.
For 1. I can show the ---> implication. If they're equivalent there exists so if the sequence converges in then it must converge in . Analogously for the other way around.
For 2. I know that is complete with the Supremum norm. Since is cts. we can rewrite this as:
where . So the new norm differs from the sup norm by a constant and then somehow show that they are equivalent. Then using a lemma: two norms are equivalent iff the normed spaces associated with each are both Banach finish it off.