Is there a generalization of matrices that allows uncountably many entries? For example, the matrix could have finitely many rows and columns, but each row/column has uncountably many elements and you can do the standard matrix multiplication by taking care to match up the entries with corresponding pairs of real number indices. 
Do such objects exist and has there been any work on them?
Does 
 A: Two such generalizations come to mind:  integral operators defined by a "kernel"  $T(f)(x) = \int K(x, y)\ f(y) dy$.  Such operators compose by convolving kernels $\int K_1(x, y) K_2(y, z) dy$, which is evidently a continuous generalization of matrix multiplication.   The second generalization is less obviously a direct generalization, but here it is:  elements in an arbitrary von Neumann factor of type II$_1$ can be regarded as continuous generalizations of finite matrices.
But you should abandon the idea of "finitely many rows and columns" and think "continuously indexed rows and columns" instead.
A: Infinite matrix rings are pretty interesting. I know several interesting examples.
The simplest is the ring of column-finite matrices over a field $F$, which is isomorphic to the ring of linear transformations of an infinite dimensional $F$ vector space. here are some of its properties. There is also, of course, the ring of row-finite matrices, and the row-and-column finite matrix rings. You could also consider the matrices with finitely many nonzero entries, and generate the ring containing those and the identity matrix.
There is also a close-knit cluster of three examples due to Bergman that all live inside infinite matrix rings. Recently, K. C. O'Meara described an algebra "containing" these examples which makes explaining them a lot easier. Here is the description of that algebra. You can find links to the Bergman examples at that link as well.
Personally, I'm curious about infinite upper-triangular matrix rings. I asked a question about a ring like that but haven't gotten any feedback on it.
