Approaches to teaching and learning analysis

Question

I found that studying linear algebra by getting into vector spaces and linear transformations first made things very easy. This is the approach Halmos or Axler, just to name a few, take.

IMHO, the alternative of going through solving systems of linear equations beforehand, obscures the topic.

What would be equivalent, slightly abstract approaches to study analysis for a freshman? A bit of topology and metric spaces, like Rudin? Multivariate calculus with differential forms, like Hubbard & Hubbard?

Answer 1

Certainly topology and metric spaces make a nice abstract (but not too abstract) intro to some of the fundamental concepts of analysis, basically continuity. But that's only one part of analysis.

Simmons Introduction to Topology and Modern Analysis is a classic that, starting with metric spaces, will take you all the way to the Gelfand-Naimark theorem on Banach algebras, one of the pinnacles of 20th century analysis. Spivak's Calculus on Manifolds is a slim volume that remains one of the best first encounters with differential forms and Stoke's theorem (that is, to someone inclined to abstraction, at the freshman level). I happen to like Lang's Analysis.

Answer 2

On the first two paragraphs of your question, I think that you are right: linear algebra first may give you one more tool to face and understand further topics in calculus/analysis.

On the other hand, another keyword you used is Topology: I think this latter, more than others, can give you a more general point of view than watching things "in $\mathbb{R}$" or "in $\mathbb{C}$".

For example, in italian degree courses in engineering we have plenty of "pure" maths subject (which is good) but not much topology (that I only studied later in BSc Maths, which is bad I think). I would add such subject to BEng courses, because I mentioned and heard about "open intervals" for years, ignoring the more general notion of open set (and this is not optimizing, in my "student's" point of view).

To make it as clear as I can, thinking of it years later, I would have preferred some more notions and background in general topology, before solving exercises about Euclidean topology of $\mathbb{R}^n$.