Let and be the -norm and the -norm on the space of continuous real valued functions on the closed interval : explicitly

, .

(a) Prove that for all (1). Hint: Remember the Cauchy-Schwartz inequality for integrals.

(b) For , define by

Caculate and .

So the thing is that I've been ill for over a week now and couldn't go to lectures resulting in me not having lecture notes and no notion of norms.

I have found hat Cauchy-Schwartz states that for two real integrable functions in an interval [a,b]. I don't really know how to use this to prove (1).

As for (b) I am confused by the sequence of functions which I don't really know how to interpret (what is it geometrically?) or plug into , .

Now, if someone could just start me off that would be great!