What pure mathematics foundations should an applied mathematician have? I'm studying mathematics, with some statistics also, and I've always chosen applied courses. I'm getting to the point where I'm studying 3rd year undergraduate to graduate level material.
My first problem is that I have the feeling that some of the time I'm just hacking, without really being sure of why I'm doing what I'm doing or whether it's even reasonable. For example with Taylor series expansions or limits I'm not really comfortable about convergence or discarding smaller terms. 
Moreover a bunch of techniques seem to crop up over and over in different guises, decomposition of a function into basis functions being a case in point -- I only noticed this was a common theme the second or third time it came up however. I keep hearing terms like subspaces but don't know what they mean.
I'm starting to think that unless I want to just be a hacker there are some fundamentals from pure maths I ought to look at. Limits being one. Vector spaces, subspaces and bases being another. An example of something that wouldn't be relevant is number theory. Problem is I don't know what I don't know so it's hard to make a list of what I should know!
My question is, what areas should an applied mathematician know at the end of a typical undergraduate course in order to have solid foundations (or at least a good intuition) for using the applied maths they have learnt?
 A: Linear algebra, for sure. Real analysis and "advanced calculus" also for sure. These concepts form the basis for numerical analysis, which is critically important for many applied fields. Even if a mathematician isn't directly working on numerical problems, there's typically a desire to develop concepts in a way that they are numerically solvable.
Probability is another field. Uncertainty quantification is hugely important in applied fields, and the interplay between probability, statistics, linear algebra, and dynamical systems cannot be overstated.
Lately, I am of the personal opinion that general algebra and things like representation and category theory are underutilized in applied fields. I have no formal basis for this, aside from my everyday work of research in engineering fields. A great many (i.e. almost all) papers completely ignore abstract formulations of the problems at hand, and they go through extreme procedural distortions to realize only moderate gains in solvability, accuracy, or computability... and so often the example still fail to represent real-world problems of interest in any meaningful way.
