I am trying to find out the essence of what a determinant is. Besides, in finite dimensions, determinant is the kind of numerical invariant that determines the invertibility of a linear operator, but what about infinite case? Is there a similar invariant? Why if not?



Is there an analog of determinant for linear operators in infinite dimensions as that of finite dimensions?
We can define the Fredholm determinant on operators on Hilbert space which differ from the identity by a trace-class operator. This satisfies 
$$\det(\exp(T)) = \exp(\text{Tr}(T))$$ for trace-class operators $T$.
See <http://en.wikipedia.org/wiki/Fredholm_determinant>


