A norm is what determines if a space is complete. Completeness is a property of a metric, every norm induces a metric given by . Now if you change a metric (or a norm), the space may not be complete. Take the metric on (which is complete under the Euclidian metric). Under this metric the sequence of natural numbers is Cauchy, but it coverges to 0 which is not in the set.(b) Prove that is complete in this norm.
Now I know that completeness of a space is equivalent to every Cauchy sequence in the space being convergent.
So let in be a sequence of functions and suppose that is Cauchy.
So for every there is an N such that .
Need to show that is convergent.
Now this is where I get confused. How does the norm relate to any of this? What does it mean for a space to be complete in a norm?
As for the solution, what can you say about and . What happens to Cauchy sequences in ?