It would be convenient, for example, if we could define a multiplicative linear functional on by , taking the limit of f along the positive real axis as r increases to 1. But unfortunately this limit may not exist. There is no easy way to specify points in the corona. However, you can see that such points must exist, as follows.
The theory of commutative unital Banach algebras tells us that there is a topology on the set of multiplicative linear functionals that makes this set into a compact space. For the point evaluations on the algebra , this topology coincides with the usual topology of . But the open unit disk is not compact, so there must be additional multiplicative linear functionals that form a 'completion' of this set. However, the completion is not the usual one under which the completion of the open disk is the closed disk. Its structure is very much more complicated.