# Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis

Greetings,

I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar transitional-curriculum courses would be interesting as well.

The context is that we are creating a 2nd year analysis course which would be our students 1st exposure to analysis. This is to be followed up by a 3rd year analysis course which would be something of a "rigorous multi-variable calculus course". Next year the 3rd year course is going to be offered but the students will not have the 2nd year course as background (in future years the 2nd year analysis course will be a prerequisite).

What they will have is a fairly extensive "service calculus" background, consisting of four courses: a more or less standard 2-course 1st year single-variable calculus sequence (text is Edwards and Penney), plus a 2nd year multi-variable calculus course (also Edwards and Penney - this is a standardish calculus in $\mathbb R^n$ for $n \leq 3$ text). They follow that up with a a 3rd year course on multi-variable calculus in $\mathbb R^n$, the text is Folland. This course also covers some material that is traditionally taught in analysis classes, things like uniform convergence, Fourier transforms and such. But very little time is spent fussing about with open subsets of $\mathbb R^n$, they don't get to see bump functions, what a function is isn't discussed in much detail, what numbers are isn't dwelled on (not even axiomatically).

So, that's the setup. It can be safely assumed these students are motivated to study analysis, as they're taking this course to transition into our upper-level analysis courses. But I can't do too much too fast. And I don't want to bore them. So things of importance for this course to dwell on are things like what numbers are (at least axiomatically), maybe even a bit of set theory, completeness, fussy details about continuous functions like when extreme values exist, the various formulations of continuity for functions on open subsets of $\mathbb R^n$. Bump functions. And fussy details from multi-variable calculus, such as the inverse and implicit function theorem.

Are there texts out there that are designed for situations like this?

One initial inclination would be to supplement something like Hubbard's calculus text with some additional foundational notes. I suppose there are many standard intro analysis courses but some of these might be oddly paced for this group of students. I presume it's unlikely anyone has written a text for just this situation but who knows?

Thanks in advance for your comments.

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## 1 Answer

If you want a reference that will not bore them, supplement the main text with "Metric Spaces: Iteration and Application" by Victor Bryant. The book is short and it shows in several contexts how the concept of a fixed-point property, via iteration, can be used to solve worthwhile problems. It is very nicely written and Bryant makes a real effort to motivate the material and explain where things are going and why.

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I think you've got a sense for where I'm going with this. They're coming from a souped-up service course sequence so it only makes sense to approach analysis from a rather applied point of view, having applied-like problems dragging them to analytical considerations. I'll look up your reference, thanks. –  Ryan Budney Mar 10 '10 at 22:21