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.

 

[LudusRex]

Real Analysis is a branch of mathematics that focuses on the study of real numbers and their properties. It is a fundamental subject in mathematics, and it plays a crucial role in many other areas, such as calculus, probability, and differential equations. In simple terms, real analysis is the study of the behavior of real-valued functions and their limits, continuity, differentiation, and integration.

To answer your question, real analysis is not just limited to a $\delta$-$\epsilon$ approach to calculus or the theory of measure and integration. It is a broad subject that encompasses both of these topics and much more. Real analysis also includes topics such as sequences and series, metric spaces, topology, and functional analysis. The $\delta$-$\epsilon$ approach is a specific technique used in real analysis to prove the convergence of a sequence or a series. Similarly, the theory of measure and integration is a powerful tool used in real analysis to study the properties of real-valued functions.

In summary, real analysis is a multifaceted subject that covers a wide range of topics and techniques. It is an essential tool for understanding the behavior of real-valued functions and their applications in various fields of mathematics.