Sobolev spaces -- I think they are pretty neat -- I'll tell you 'bout 'em.
In developing a theory of partial differential equations, we must choose a setting for solutions. For example, we could look for smooth, or analytic solutions, or solutions in -- the space of functions which are times continuously differentiable. Introducing a notion of weak solution allows us to work in spaces with more structure. Typical settings for these weak solutions are the Sobolev spaces . We will discuss the definitions of weak solutions and Sobolev spaces, as well as applications of the Riesz representation theorem and Banach contraction theorem to PDE through Sobolev spaces.
Sobolov Spaces and the Question "What is a Solution to a Partial Differential Equation?"