A linear operator T on a vector space X is a function T: X → X that satisfies:
Let X = C[0, 1] and define ||.||: X → ℝ by ||f|| = ∫[0, 1] |f(x)| dx. Show that ||.|| is a norm.
A particularly tricky section of Chapter 2 deals with spaces that are not complete. A classic problem involves proving that the space of polynomials on $[0,1]$ is not complete.