MHB What exactly do you call Real Analysis?

[ModusPonens]

Hello

I'm curious to know what exactly do americans call real analysis. Is it a $\delta$ $\epsilon$ approach to calculus? Or is it the theory of measure and integration, consisting mostly of the Lebesgue integral?

EDIT: I didn't want to disrupt the topic on the motivation letter for graduate school.

 

[Ackbach]

Hello

I'm curious to know what exactly do americans call real analysis. Is it a $\delta$ $\epsilon$ approach to calculus? Or is it the theory of measure and integration, consisting mostly of the Lebesgue integral?

EDIT: I didn't want to disrupt the topic on the motivation letter for graduate school.

The answer is "yes". Real analysis at the undergraduate level in the US is typically $\delta-\epsilon$ proofs of the big theorems in calculus, and plenty of sequences, both of numbers and functions. The stereotypical book is Rudin's Principles of Mathematical Analysis. Graduate-level real analysis is measure and integration, including Lebesgue and generalized measure integrals (the proofs are all the same, so a Caratheodory approach, e.g., does them more or less simultaneously for more generality). Real analysis does not typically include significant functional analysis, although functional analysis does depend on real analysis. The stereotypical book here is Royden's Real Analysis.

 

[ModusPonens]

But do you have a separate course for calculus and undergrad real analysis?

 

[Ackbach]

Yes, we do. We usually have three semesters of Calculus (roughly differential, integral, and multivariable), followed by an introduction to differential equations. That finishes up the sophomore year, although many colleges also offer linear algebra and discrete mathematics in the sophomore year as well.

Some colleges offer what they call advanced calculus in the junior year, which can be anything from full-blown real analysis to heavily applied multivariable calculus. Real analysis, the full $\delta-\epsilon$ proof course, is usually a senior-level course.