What makes a set of objects a vector space? You will no doubt want to refer to notes and the text, but I'd like you to summarize it for starters. If you have identified a vector space, for example the so-called L2 space of square- integrable functions, what advantages does it give you? Why do non-linear boundary value problems, i.e., problems with non-linear ODE's or PDE's, fall outside the field of linear spaces? If you can, show by example what the problem is in practice.