I am strugling with this exercise:
Let . Prove that in
The solution given to me reads:
Since the given space is the dual of
Thinking of the as linear functionals with domain in
Prove that for all
It is enough to prove it holds for in a set whose span is dense in , like the .
That is straightforward since I can easily integrate sin(nt) over [a,b]and notice it converges to 0, but I don't understand why/ I am not convinced that it is enough to do so. I have read elsewhere that " a sequence of continuous functions on a metric space that converges pointwise on a dense subset need not converge pointwise on the full space.",
So why does it work in this particular case? Can you prove it?
I have the following proven propositions, but none of them are exactly what I need, since this problem deals with weak * convergence instead
Proposition 1: X: Banach space, , such that . Let , bounded such that . Then such that
which was actually used to prove:
Proposition 2: X: Banach space, iff is bounded and 'such that