TITLE: Is there an analog of determinant for linear operators in infinite dimensions as that of finite dimensions?
QUESTION [11 upvotes]: 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?

REPLY [3 votes]: The answers above point out that one cannot define a determinant in a meaningful way on the algebra of bounded operators on a Banach space, unless finite-dimensional.  However, this does not preclude the possibility of doing this for suitable subclasses and this is precisely what Alexander Grothendieck did  in his celebrated (amongst functional analysts)
article "Théorie de Fredholm" (Bull. Soc. Math. vol. 84---freely available online).
This is one of those articles which changed the face of mathematics forever.  The closely related question of which operators have a  trace has been investigated in great detail, by, for example,  Albrecht  Pietsch and Hermann König.