TITLE: Can linear maps between infinite-dimensional spaces be represented as matrices?
QUESTION [21 upvotes]: Any linear map between two finite-dimensional vector spaces can be represented as a matrix under the bases of the two spaces. 
But if one or all of the vector spaces is infinite dimensional, is the linear map still represented as a matrix under their bases?
If there is matrix of infinite dimension, what is it used for if not used as a representation of a linear map between vector spaces?
Thanks and regards!

REPLY [12 votes]: What, however, must be understood that the role of matrices when one works with linear operators of infinite-dimensional vector spaces is a (very) marginal one. Familiar techniques such as the use of determinants, traces etc. no longer work. For instance, any determinant-like function on $\mathrm{GL}(V)$ where $V$ is an infinite-dimensional vector space over a field is necessarily the trivial one, since any element of $\mathrm{GL}(V)$ is a product of commutators (the group $\mathrm{GL}(V)$ is perfect, in other words; a result by Alex Rosenberg of 1958.)