Function spaces and compactness

It is useful to treat real-valued functions (or complex-valued functions, or vector space-valued functions) as elements of a vector space, so that the tools from linear algebra can be applied. Given a set X one may consider the vector space R of all real-valued functions with domain X. If X is finite, say with n elements, then this is just the familiar vector space R. The more interesting examples are when X is infinite, and so R is infinite-dimensional. We will focus on the case X = [0, 1], which is reasonably representative. Generally speaking, the functions [0, 1] → R that arise from some application are not entirely arbitrary, but have some degree of regularity – maybe they are continuous, or piecewise continuous, or measurable, or integrable, etc. It turns out that the vector space R is “too large” for many applications, and that it is more suitable to consider a smaller space, whose elements are functions with some extra properties. We will consider some of the ways to do this, paying particular attention to how those choices let us recover certain properties of R that involve extra structure beyond that of the vector space itself: