Analysis Preliminary Exam Workshop: Distributions and Sobolev Spaces

A distribution is a linear functional on a space of test functions. Distributions include all locally integrable functions and have derivatives of all orders (great for linear problems) but cannot be multiplied in any natural way (not so great for nonlinear problems). One can use many different spaces of test functions. We will consider distributions on the space D(Ω) of smooth compactly supported test functions where Ω ⊂ R is an open set (which we call simply “distributions”), distributions on the Schwartz space S(R) of smooth, rapidly decreasing functions (“tempered distributions”), and distributions on the space D(T) of smooth periodic functions (“periodic distributions”).