(i) Suppose that the real-valued weight function is defined, continuous, positive and integrable on the interval . Then, for any function ,
(ii) Given any two positive numbers (however small) and (however large), there exists a function such that
(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.