Orthonormal basis in Hilbert space

Hi,

The first part of a problem was to show that if Y is a dense subspace of a separable hilbert space H, then H has an orthonormal basis consisting of elements in Y.

I was able to do this.

The second part of the problem is use this result to prove that if , there is an orthonormal basis of continuous bounded functions such that .

I can see that if we define the linear functional , with domain the set of continuous functions of compact support intersected with , by , then we want to take to be and in the above result. It suffices to show is dense in since is dense in . I know that this is equivalent to being discontinuous (ie. unbounded), but I'm getting stuck on using that isn't in to show is unbounded.

Is this the right idea? If so, how can we use that is not in to show that is unbounded?

Many thanks.

Edit: I hadn't solved the problem, but thought the above was too wordy so wished to replace it.