Let be the space of differentiable real valued functions such that the derivative is continuous. Given define by

.

(a) Prove that is a norm on .

Ok, so I have absolutely no idea how to do this. What defines a norm on a space of functions? How does one show that something is a norm?

(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?