What text are you using?
I am looking for an example of a state (positive linear functional of norm 1) on the space of continuous functions on the torus .
My first guess would be something like
with norm
with . However, if we take the function to map any element of the torus identically to 1 the above definition would imply that
so this idea does not work. Any ideas?
I don't think that will work. It will be fine for that particular function, but we can just as well consider a function that maps any element of the torus to say 2, then the integral will have twice that value. I think I have an idea how to construct such a state, will post it if it works.