Hi.

Problem:

(i)Suppose that the real-valued weight function is defined, continuous, positive and integrable on the interval . Then, for any function ,

,

where

.

(ii)Given any two positive numbers (however small) and (however large), there exists a function such that

,

.

Attempt:

(i)By using integration by parts I get,

To me that looks good, but I've been mistaken before

(ii)I do not see how I can prove existence of the function. This somehow reminds me of the dirac delta function, as I can make my function as pointy as I want...

The book continues by saying that the Lemma indicates that the norms are not equivalent as they would be if this was a finite-dimensional linear space and we were dealing with vectors. That is if and are vector norms on then

Does this mean that if we had , then we could use the two norms interchangeably? I do not immediately see how the Lemma indicates that this is not so.

Thanks!