Thread: State on classical space

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?

What text are you using?

Following your ideas, |w(f)| < 4(pi)^2 ||f|| where the last norm is the one in your space so what about v(f)=w(f)/4(pi)^2?

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.

I was assuming there was a typo in your definition of the norm of the functional since by linearity that quantity will never be bounded. So I assumed you meant and for this definition my example does work.